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THE 

STRENGTH OF MATERIALS 



Sontion: C. J. CLAY and SONS, 

CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, 

AVE MARIA LANE. 

©lasgoto: 263, ARGYLE STREET. 




iUipjig: F. A. BROCKHAUS. 
iforfe: THE MACMILLAN COMPANY. 
JSombag: E. SEYMOUR HALE. 



THE 

STRENGTH OF MATERIALS 



BY 



QJ 



Jf A: EWING, M.A., B.Sc., F.RS., M.Inst.C.E. 

PROFESSOR OF MECHANISM AND APPLIED MECHANICS IN 

THE UNIVERSITY OF CAMBRIDGE, 

FELLOW OF KING'S COLLEGE, CAMBRIDGE. 



> • ■ , 

. i , , 

• ' > A -. ' • •, , 

' • 4 9*9 



CAMBRIDGE : 
AT THE UNIVERSITY PRESS. 

1899 

I .(// Rights reserved. I 









. 



ft f- 



€zrr.'z-:':~: 



HSTE.I) BY J. A>"D C. F. CLAY. 

at the xryrvEP.?rrY press. 



PEEFACE. 

|~N modern schools of Engineering a student acquires his know- 
ledge of the Strength of Materials and of its application in 
design, partly by hearing lectures, partly by making experiments 
in the laboratory, and partly by working out examples in the 
drawing-office. The present treatise is an attempt to set forth 
briefly a lecture-room treatment of the subject, which to be effec- 
tive must be supplemented by laboratory and drawing-office work. 
Indications are also given of some laboratory experiments in 
elasticity, and a number of pieces of apparatus are described 
which have proved serviceable at Cambridge. 

I am indebted to Messrs A. and C. Black for permission to use 
the substance of the article " Strength of Materials " which I 
wrote for the Ninth Edition of the Encyclopaedia Britannica. 
Also to Professor Unwin, and his publishers Messrs Longmans, 
for the illustrations on page 74, which are taken from his valuable 
Treatise on the Testing of Materials. To Mr T. Peel of Magdalene 
College I owe much for his kindness and care in reading the proofs 
of these sheets. 

J. A. EWING. 

Engineering Laboratory, Cambridge. 
October, 1899. 



k. s. m. 



CONTENTS. 



CHAPTER I. 

STRESS AND STRAIN. 

ART. PAGE 

1. Introductory 1 

2. Stress 2 

3. State of Stress 2 

4. Condition of Equilibrium 2 

5. Distribution of Stress. Intensity of Stress .... 3 

6. Normal and Tangential Stress 3 

7. State of Simple Push or Pull 4 

8. Complex states of Stress. Principal Stresses .... 4 

9. Character of the Stress in Simple Push or Pull ... 5 

10. Combination of two simple pull or push stresses in directions 

at right angles to one another 6 

11. State of Simple Shear 7 

12. Equality of Shearing Stress in two directions .... 9 

13. Fluid Stress 10 

14. Strain 10 

15. Elastic Strain and Permanent Set. Limits of Elasticity . . 10 

16. Hooke's Law 11 

17. Young's Modulus or the Stretch Modulus L2 

18. Ratio of Lateral Contraction to Longitudinal Extension in 

Simple Pull 12 

19. Strain produced by Shearing Stress. Modulus of Rigidity or 

Shear Modulus . 13 

20. Modulus of Cubic Compressibility or Bulk Modulus . . L9 

21. Relation between the Moduluses of Elasticity . . . . II 

22. Work done in produoing an Elastic Strain : Resilience . . l i 



Vlll 



CONTEXTS. 



CHAPTER II. 



RELATIONS BETWEEN THE ELASTIC CONSTANTS. 

ART. 

23. Relation between E, C, and K 

24. Relation of <r to the moduluses C and K . 

25. Other expressions of relation between the Elastic Constants 

26. Isotropic material : Equations connecting Stresses and Strains 

27. State of Simple Shear .... 

28. Volume Strain ..... 

29. Simple Strain along one axis 

30. Lateral Strain prevented in one direction 

31. Numerical Example .... 




PAGE 

16 

18 

19 

20 

21 

21 

22 

22' 

23 



CHAPTER III. 



ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 

32. Strain carried beyond the Limit of Elasticity 

33. Ultimate Strength 

34. Factor of Safety 

35. Variation of the LTltimate Strength under different modes of 

loading . 

36. Advantage of Plasticity ...... 

37. Ordinary Tensile Tests ...... 

38. Diagrams of Extension and Load in Tensile Tests 

39. Autographic Diagrams of Extension and Load . 

40. Hardening effect of rest after overstraining 

41. Breakdown and Recovery of Elasticity after overstraining 

42. Effect of Heating in facilitating Recovery of Elasticity after 

overstraining .... 

43. Influence of time in the process of testing 

44. Effects of Hardening through overstrain 

45. Annealing ...... 

46. Hardening and Tempering of Steel 

47. Contraction of section at rupture 

48. Non-elastic Extension .... 

49. Crystalline Structure of Metals . 

50. Percentage of Extension in Tensile Tests 

51. Forms of Test-Pieces .... 

52. Comparative Strength of Long and Short Rods 

53. Fracture by Tension or by Compression 

54. Inclination of Surfaces of Shear in Tension and Compression 

Tests 



24 
25 

27 

28 
28 
29 
30 
32 
33 
35 

36 
41 
42 
43 

44 
44 
45 
46 
47 
47 
49 
50 

51 



CONTENTS. 



IX 



ART. 

55. 
56. 
57. 
58. 
59. 



Fatigue of Metals .... 

Imperfection of Elasticity 
Cumulative Effect of Blows and Shocks 
Initial Internal Stress .... 
Influence of Temperature on Strength 



PAGE 

52 
54 
56 
57 
57 



CHAPTER IV. 



THE TESTING OF MATERIALS. 

60. Testing Machines 

61. Single-Lever Machine with vertically placed Test-Piece 

62. Calibration of Vertical Machines .... 

63. Other Testing Machines using Weights and Levers . 

64. Other Testing Machines. Diaphragm Machines 

65. Testing Machines for special purposes 

66. Attachment of the specimens in tensile tests 

67. Apparatus for drawing autographic diagrams of extension and 

load 

68. Measurement of Young's Modulus by Extensometers 

69. Measurement of Young's Modulus in Wire 

70. Measurement of Young's Modulus by Bending . 

71. Measurement of the Modulus of Rigidity. Static Method 

72. Measurement of the Modulus of Rigidity. Kinetic Method by 

Torsional Oscillations . 

73. Maxwell's Needle used as Torsional Oscillator 

74. Results of Tests. Data for Cast-iron 

75. Wrought-iron 

76. Steel 



59 
59 
63 
64 
67 
69 
70 

72 
73 
81 
83 

85 

87 
88 
90 
91 
92 



CHAPTER V 



UNIFORM AND UNIFORMLY- VARYING DISTRIBUTIONS OF STRESS. 



77. Use of the Stress Figure to represent a Stress distributed 

over a Surface ..... 

78. Uniformly distributed Stress 

79. Uniformly-varying Stress .... 

80. Uniformly-varying Stress forming a Couple 

81. Analysis of any Uniformly-varying Stress into a uniform Stress 

and a couple . . . ... 

82. Extent to which Stress may be Non-axial without reversing 

its sign at the edge of the stressed surface 



96 
96 
99 

101 
102 
LOS 



CONTEXTS. 



AKT. 

83. 

84. 

85. 



Simple Bending 

Influence of Bending beyond the Elastic Limit on the Distri- 
bution of the Stress 

Modulus of Rupture 



PAGE 

104 

106 
107 



CHAPTER VI. 

STRESS IX BEAMS. 

86. Character of the Stress in Beams ...... 109 

87. Stress due to Bending Moment 110 

88. Particular Cases Ill 

89. Beam with Flanges and "Web 112 

90. Variation of Bending Moment and Shearing Force from point 

to point along a Beam. Diagrams of Bending Moment 

and Shearing Force .' 113 

91. Graphic method of finding Bending Moments . . . . 120 

92. Relation between the Bending Moment Diagram and the 

Funicular Polygon for the same system of Loads . . 120 

93. Connection between Bending Moment and Shearing Force . 122 

94. Bending Moment and Shearing Force due to Moving Loads . 123 

95. Distribution of Shearing Stress over the Section of a Beam . 124 

96. Principal Stresses in a Beam 126 



CHAPTER TIL 



DEFLECTIOX OF BEAMS: COXTIXUOUS BEAMS. 

97. Curvature due to Bending Moment 129 

98. Condition of uniform curvature 130 

99. Relation of Curvature, Slope and Deflection .... 131 

100. Examples of Slope and Deflection in Beams and Cantilevers 131 

101. Deflection of a uniform beam under a single load placed 

anywhere .......... 134 

102. Transverse Bending. Anticlastic curvature .... 135 

103. Resilience of a Beam 136 

104. Graphic method in the treatment of Deflection . . . 138 

105. Additional Deflection due to Shearing 141 

106. Continuous Beams 143 

107. Theorem of Three Moments 146 

108. Advantages of Continuous Beams 148 

109. Combination of Cantilever with Beams 149 

110. Encastre Beam 150 



CONTENTS. 



XI 



CHAPTER VIII. 

FRAMES. 

ART. PAGE 

111. Frames 154 

112. Perfect, Imperfect, and Redundant Frames .... 155 

113. Method of Sections 156 

114. Graphic Process. Method of Reciprocal Figures . . . 157 

115. Examples of the Method of Reciprocal Figures . . . 159 

116. Use of the Funicular Polygon in finding the Reactions at 

the Supports . 162 

117. Special Cases 164 

118. Use of Semi-members in Bridge Frames. Counter-bracing . 166 

119. Superposed Frames 169 

120. Effects of Stiff Joints 170 



CHAPTER IX. 

STRUTS AND COLUMNS. 

121. Instability under Compression 

122. Bending of Long Columns. Euler's Theory . 

123. Fixed and Free Ends 

124. Modification of Euler's theory to meet practical conditions 

125. Values of the Constants in other materials . 

126. Struts with Lateral Load 



171 
172 

174 
176 
182 
184 



CHAPTER X. 

TORSION OF SHAFTS. 

127. Torsion of a uniform circular shaft 

128. Relation of the greatest intensity of stress to the twisting 

moment in solid and hollow circular shafts 

129. Angle of Twist in Round Shafts . 

130. Relation of Power transmitted by a Shaft to Torsional Stress 

and Angle of Twist .... 

131. Twisting combined with Bending 

132. Resilience of a Round Shaft under Torsion 

133. Torsion beyond the Elastic Limit . 

134. Spiral Springs 

135. Helix in which the obliquity is considerable 
L36. Torsion of non-circular shafts . 

137. Stability of Shafts under End Thrust and r I 

1 38. Centrifugal whirling <»f shafts . 



orsion 



187 

L88 

189 

L90 

191 
193 

191 

L96 
L96 
L99 

80] 



XII 



CONTEXTS. 



CHAPTER XL 

SHELLS AND THICK CYLINDERS. 

ART. PAGE 

139. Stress in a Thin Shell due to Internal Pressure . . . 204 

140. Longitudinal Stress in Cylinder exposed to Internal Pressure 206 

141. Spherical Shell 207 

142. Cylindrical Shell of aval section 207 

143. Thick Circular Cylinder 208 

144. Thick Cylinder exposed to External Pressure . . . 211 

145. L'se of Initial Internal Stress in strengthening a thick tube 211 

146. Stress in a Revolving Ring 214 

147. Stress in a Revolving Disc 215 



CHAPTER XII. 



HANGING CHAINS A>"D ARCHED RIBS. 



148. Loaded Chain 

149. Parabolic Chains ..... 

150. Common Catenary : Uniform Chain loaded 

weight 

151. Suspension Bridge with Stiffening Girder 

152. Inverted Chain. The Arch 

153. Arched Rib 

154. Rib hinged at the ends and centre 

155. Rib hinged at the ends only . 

156. Rib fixed at the ends .... 



with its 



own 



218 
220 

221 
223 
228 
230 
231 
233 
236 



APPENDIX. 



Table I. Strength to resist Tension 

Table II. Strength to resist Crushing . 

Table III. Strength to resist Shearing 

Table IV. Moduluses of Elasticity . 

Table V. Approximate "Weights of Materials 



239 
241 
241 

241 
242 



Index 



243 



CHAPTER I. 



STRESS AND STRAIN. 



1. Introductory. The term " Strength of Materials " is 
used in a somewhat wide sense to name that part of the Theory 
of Engineering which deals with the nature and effects of stresses 
in the several parts of engineering structures. When a structure 
is loaded, that is to say, when forces of any kind are applied to it, 
the applied forces cause the parts of the structure to be stressed 
in various ways. Unless the parts are severally strong enough 
and stiff enough to bear these stresses the structure fails. To 
determine beforehand what loads the structure will bear safely, 
or conversely to design a structure which will be safe under a 
given set of loads, requires two things. We must be able to 
analyse the stresses in the various parts of the structure and by 
determining their relation to the applied loads to calculate their 
amounts. And further, we must know by experiment the pro- 
perties of the materials which form the structure, both as to 
strength and as to stiffness, in order to judge after the action of 
the load has been analysed, what dimensions should be given to 
the parts to make them individually safe, and to prevent the 
structures as a whole from being unduly strained out of shape. 

Hence the subject has two sides. On one hand, it is experi- 
mental and deals with the properties which materials arc found 
to possess as to strength and elasticity. On the other, it Is 
mathematical and discusses the kinds of stress to which the pieces 
of structures are subject, and also the changes of form which occur 
in consequence of the fact that all materials are more or less 
clastic. 

E. S. M. 1 



2 STRESS AND STRAIN. 

2. Stress. Stress is the mutual action between two bodies, 
or between two parts of a body, whereby each of the two exerts a 
force upon the other. 

Thus when a stone lies upon the ground there is at the surface 
of contact a stress, one aspect of which is the force which the stone 
exerts upon the ground, pushing the ground downwards, and the 
other aspect is the equal force directed upwards which the ground 
exerts upon the stone. Newton's " Third Law," that " action and 
reaction are equal and opposite," may be paraphrased by the state- 
ment that every force is one aspect of a stress. A stress may exist 
between two separate bodies, or between portions of a single body 
separated only by an imaginary surface of division. In a tie-rod, 
for instance, which is bearing a pull there is a stress between the 
tw T o parts into which the rod may be imagined to be divided by 
any plane of cross-section : each part exerts a pull upon the other 
part across the plane. 

3. State of Stress. A body is said to be in a state of stress 
when there is stress between the two parts which lie on opposite 
sides of any imaginary dividing surface. Thus the tie-rod of the 
last example is a body in a state of stress because there is a pull 
between the parts into which the rod is cut by suciy imaginary 
surface of cross-section. 

A pillar or block supporting a weight is in a state of stress 
because at any cross-section the part above the section pushes 
down against the part below, and the part below pushes up against 
the part above. A plate of metal that is being cut in a shearing 
machine is in a state of stress, because at the plane which is about 
to give way by shearing the portion of metal on either side is 
tending to drag the portion on the other side with a force in that 
plane. 

4. Condition of Equilibrium. The kind and amount of 
stress which exists over any surface within a body at rest is in 
general to be determined by considering that if the body is con- 
ceived to be divided into two parts A and B by the surface in 
question the force which A exerts upon B across the surface 
must equilibrate all the other forces which act on B, namely, the 
loads or external forces which are applied to it, including its 
weight and any forces which are exerted on it by its supports. 
Similarly the forces which act on A must when taken together be 



STRESS AND STRAIN. 



3 



in equilibrium, and the forces exerted by B upon A must balance 
the other forces which act on A. Thus the stress between A and 
B may be investigated by considering the equilibrium of either 
A or B. 

5. Distribution of Stress. Intensity of Stress. A stress 
acting at a surface is distributed over it, each square inch or other 
portion of the surface bearing so much. The distribution may or 
may not be uniform. If it is uniform every square inch or other 
unit of area in the surface bears the same amount of the stress as 
every other. The intensity of stress, by which is meant the 
amount of stress per unit of area, is in that case found by dividing 
the whole stress by the whole area. Thus if a stress of P tons is 
uniformly distributed over a surface of S square inches, the inten- 
sity p in tons per square inch is given by the equation 

P 



P 



8' 



When the distribution is not uniform there is still a definite 

intensity of stress at any point in the surface, the value of 

which is 

BP 

BS 

where BS is an indefinitely small area surrounding the point and 

BP is the stress acting on that small area. For practical purposes 

the intensity of a stress is usually expressed in tons weight per 

square inch, lbs. weight per square inch, or kilogrammes weight 

per square millimetre or per square centimetre*. 

6. Normal and Tangential Stress. When a solid body is 
in a state of stress the direction of 
the stress at any imaginary surface of 
division may have any inclination to 
the surface ; it may be normal to the 
surface, or tangential to it, or oblique. 
A stress the direction of which is 
oblique to the surface is most con- 
veniently treated by resolving it into 
normal and tangential components. a 
Thus if p r (fig. 1) be the intensity 

* One ton per square inch = 2240 lbs. per square inch 157*2 kilos per Bquare 

centimetre. 

1— J 





Fig. 1. 



4 STRESS AND STRAIN. 

of stress on the surface AB, the direction of the stress making 
an angle 6 with the normal to the surface, this oblique stress 
is equivalent to a normal stress p n together 'with a tangential 
stress p t , the intensity of the normal component being 

p n =p r cos0 
and that of the tangential component 

p t =p r san0. 
Xormal stress may consist either of push (compressive stress) or 
of pull (tensile stress) : if a stress of pull be taken as positive a 
stress of push will be negative. Xormal stress tends to make the 
portions which lie on the two sides of the surface directly recede 
from each other if it is positive, or directly approach each other if 
it is negative. 

Stress which is tangential to the surface is often called Shear- 
ing Stress. It tends to make the material on one side of the 
surface slide past the material on the other. 

7. State of Simple Push or Pull. The simplest possible 
state of stress is that of a short pillar or block compressed by 
opposite forces applied at its ends, or that of stretched rope or 
other tie. In these cases the stress is wholly in one direction. 
These cases may be distinguished as simple push and simple pull. 
In them there is no stress on planes parallel to the direction of 
the applied forces. 

8. Complex states of Stress. Principal Stresses. A 

more complex state of stress occurs if the block (which for 
simplicity of statement we may assume to have a rectangular 
cross-section) is compressed or extended by forces applied to a 
pair of opposite sides, as well as by forces applied to its ends — 
that is to say, if two simple push or pull stresses in different 
directions act together. A still more complex state occurs if a 
third push or pull be applied to the remaining pair of sides. It 
may be shown that any state of stress which can possibly exist 
at any point of a body may be produced by the joint action of 
three simple stresses of push or pull in three suitably chosen 
directions at right angles to each other. These three are called 
principal stresses.^ and their directions are called the axes of 
principal stress. The axes of principal stress have the important 
property that the intensity of stress along one of them is greater, 
and along another is less, than in any other direction. These 



STRESS AND STRAIN. 



are called respectively the axes of greatest and least principal 
stress. We shall have examples later on of more or less complex 
modes of stress, in which it will be important to calculate the 
principal stresses, since the greatest principal stress measures the 
greatest intensity which the material has to bear. 

9. Character of the Stress in Simple Push or Pull. 

Returning now to the state of stress which is produced by a single 

simple pull or push, let AB (fig. 2) be a 

portion of a tie or strut which is being 

pulled or pushed in the direction of the 

axis AB with a total stress P uniformly 

distributed. On any plane section CD 

taken at right angles to the axis there 

is a normal pull or push of intensity 

P 

p = -~ , S being the area of the normal 

cross section. On such a plane there is 
no tangential stress. But on any plane 
EF whose normal is inclined to the axis, 
the stress is still in the direction of the 
axis, and is therefore oblique to the plane 
EF. Hence on such a plane there is 
tangential as well as normal stress. Let 
6 be the angle which the normal to the 
inclined section EF makes with the direc- 
tion AB along which the stress acts. The 
area of the inclined section is 

S 



S' = 



cos 6' 



b: 
Fig. 2. 



The total stress P acting on this area may be resolved into the 
normal component 

P» = P cos 6, 

and the tangential component 

P t = P sin 6. 
Dividing these by the area S' over which they act we find the 
intensities of the components as follows: — The intensity ot' the 
normal stress on EF is 

Peostf P , ... 

/>„= .„ = s , CO8 9 0=2>CO8*0, 



STEESS AXD STRAIN. 



and the intensity of the tangential or shearing- stress on EF is 



P sin 
Pt = — ™ — =psin 



a a p sin 20 
6 cos 6 = r — — 



This intensity of tangential or shearing stress reaches a maximum 
when the surface EF has an inclination of 45° to the direction of 
the pull, for sin 20 has then its greatest possible value. It is clear 
that any surface having this inclination, whether plane or not, will 
be a surface of maximum shearing stress, and the intensity of the 
shearing stress upon it will be 

• A - « - - P 

max. p t = p sm 4o~ cos 4o —- . 

This production of shearing stress, on inclined surfaces, by the 
application of simple pull or push finds an important illustration 
in the testing of materials. When a bar is pulled asunder, or a 
block is crushed by pressure applied to two opposite forces, it 
frequently happens that yielding takes place wholly or in part by 
shearing on surfaces inclined to the direction of the pull or the 
thrust. 

10. Combination of two simple pull or push stresses in 
directions at right angles to one another. Suppose that in 
addition to the simple pull or push of fig. 2 there is a second pull 
or push stress acting at right angles to the first, as iu fig. 3. On 
any surface EF inclined as in the figure there will be a stress the 




Fig. 3. 



STRESS AND STRAIN. 7 

normal and tangential components of which are readily found as 
follows. Let p X) p y be the intensities of the stresses which P x and 
P y respectively produce on planes perpendicular to their own 
directions, and let the plane EF be inclined so that its normal 
makes an angle 6 with the direction of P x and 90° + 6 with the 
direction of P y . Then by summing the effects due to P x and P y 
separately we have, for the intensity of normal stress on EF, 

Pn =Px cos 2 6 +p y cos2 ( -£ + 



2 

=p x cos 2 + p y sin 2 6. 
Again, for the intensity of tangential stress on EF, 

p t =p x sin 6 cos 0+p y sin / y + 0) cos I ~ + 6 I , 

= {p x —Py) sm cos 0- 

This tangential stress becomes a maximum as before when the 
inclination of the surface is 45°, and its value then is 



Max. p t = 



Px p y 



2 
The normal stress on the same surface, inclined at 45°, is 

P*+Pv 
2 ' 

11. State of Simple Shear. A special case of great import- 
ance in practice occurs when the two simple stresses of § 10 are equal 
in intensity but opposite in sign : in other words, when one is a push 
and the other is an equal pull. When this happens there is no 
normal stress on a plane inclined at 45° to the two directions, for the 
normal component of the pull is equal and opposite to that of the 

push. The expression ^-~ :y vanishes when p, t = —p x - In other 

words, there is nothing but tangential or shearing stress on the 
two planes which are inclined at 45° to the axes along which the 
pull and push act. And the intensity of the shearing stress on 
each of these planes, namely 

Pz-P,, 

2 ' 

is numerically equal to p x or to p yt 

This is called a state of simple shearing Stress, <»r mere briefly 

a state of simple shear. It may be described as a state in which 



8 



STRESS AND STRAIN. 



there are two principal stresses only, one equal and opposite to 
the other. These two principal stresses give rise to a stress 
which is wholly tangential on the two planes inclined at 45 = to 
the axes of principal stress, and the intensity of the tangential 
stress on each of these planes is equal to the intensity of either of 
the principal stresses. 

The state of simple shear may also be arrived at in another 
way. Let a cubical block or an elementary cubical part of any 
solid body (fig. 4) have tangential 
stresses QQ applied to one pair of 
opposite faces, A and B, and equal 
tangential stresses applied to a 
second pair of faces C and D, as 
in the figure. The effect is to 
set up a state of simple shear. 
On all planes parallel to A and B 
there is nothing but tangential 
stress and the same is true of all 
planes parallel to C and D. The 
intensity of the stress on both 
systems of planes is equal through- 
out to the intensity which was applied to the face of the block. 

To see the connection between these two w r ays of specifying a 
state of simple shear we have only to consider the equilibrium of 




Fig. 4. 





the parts into which the block may be divided by ideal diagonal 
planes of section. To balance the forces QQ (fig. 5), there must 
be normal pull on the diagonal plane, the amount of which is 
P = v2Q. But the area of the surface over which P acts is greater 
than that of the surface over which Q acts in the proportion 



STRESS AND STRAIN. 



9 




Fig. 6. 



which P bears to Q, and hence the intensity of P is the same as 
the intensity of Q. 

Again, taking the other diagonal plane (fig. 6), precisely the 
same argument applies except that here 
the normal force P required for equili- 
brium is a push instead of a pull. Its 
intensity has the same value as before. 
Thus the state of simple shearing stress 
defined as consisting of two equal tan- 
gential stresses on two planes at right 
angles to one another is found to admit 
of analysis into two equal principal 
stresses, one of push and one of pull, 
acting in directions at right angles to 
one another and inclined at 45° to the directions of the shearing 
stress, just as the combination of a push with an equal pull at 
right angles to it has already been found to set up a state of 
simple shear. 

12. Equality of Shearing Stress in two directions. No 

tangential stress, whether occurring by itself or as the tangential 
component of an oblique stress, can exist in one direction without 
an equal intensity of tangential stress existing in another direc- 
tion at right angles to the first. To prove this it is sufficient to 
consider the equilibrium of an indefinitely small cube (fig. 7), with 
one pair of sides parallel to the direc- 
tion of the shearing stress. This aQ' 
stress, acting on two opposite sides, 
produces a couple which tends to 
rotate the cube. No arrangement of 
normal stresses on any of the three 
pairs of sides of the cube can balance 
this couple ; that can be done only 
by a shearing stress Q' whose direc- 2f 
tion is at right angles to the first 
stress Q, and to the surface on which 
Q acts, and whose intensity is the 
same as that of (,). The argument is equally valid whether these 
tangential stresses act alone or as the tangential components of 

oblique Stresses, and also whether there is or is not, other stress on 



5* 



YQ' 



Fig. 7. 



10 STRESS AND STRAIN. 

the remaining faces of the cube. If Q and Q' act alone we have 
the condition of simple shear described in the last paragraph. 

13. Fluid Stress. Another important case is found when 
there are three principal stresses all of the same sign and of equal 
intensity p. It may be shown that the tangential components on 
a plane inclined in any direction cancel each other. There is no 
shearing stress anywhere ; the stress on every plane is wholly 
normal and its intensity is p. This is the only state of stress that 
can exist in a mass of fluid at rest, in consequence of the fact that 
a fluid can exert no statical resistance to shear. For this reason 
the state is often briefly described as a fluid stress. 

14. Strain. Strain is the change of shape jDroduced by 
stress. If the stress is a simple longitudinal pull, the strain 
consists of lengthening in the direction of the pull, accompanied 
by contraction in both directions at right angles to the pull. If 
the stress is a simple push, the strain consists of shortening in the 
direction of the push with expansion in both directions at right 
angles to that ; the stress and the strain are then exactly the 
reverse of what they are in the case of simple pull. If the stress 
is one of simple shear, the strain consists of a distortion such as 
would be produced by the sliding of layers in the direction of 
the shearing stresses. 

15. Elastic Strain and Permanent Set. Limits of 
Elasticity. A material is elastic with regard to any applied 
stress if the strain disappears when the stress is removed. Strain 
which persists after the stress that produced it is removed is called 
permanent set. For brevity it is convenient to speak of strain 
which disappears when the stress is removed as elastic strain. 

Actual materials are in general nearly perfectly elastic with 
regard to small stresses, and very imperfectly elastic with regard 
to great stresses. In most materials, if the applied stress is less 
than a certain limit, the strain is small in amount, and disappears 
wholly or almost wholly when the stress is removed. If the 
applied stress exceeds this limit, the strain is, in general, much 
greater than before, and the chief part is found, when the stress is 
removed, to consist of permanent set. The limits of stress within 
which strain is wholly or almost w T holly elastic are called elastic 
limits or limits of elasticity. 



STRESS AND STRAIN. 11 

For any particular mode of stress the limit of elasticity is much 
more sharply defined in some materials than in others. When 
well denned it may readily be recognised in ihe testing of a 
sample from the fact that after the stress exceeds the limit of 
elasticity the strain begins to increase in a much more rapid ratio 
to the stress than before. This characteristic goes along with the 
one already mentioned, that up to the limit the strain is wholly 
or almost wholly elastic. 

16. Hooke's Law. Within the limits of elasticity the 
strain produced by a stress of any one kind is proportional to 
the stress producing it. This relation between elastic strain and 
stress was enunciated by Hooke in 1676, and is known as Hooke's 
Law. 

In applying Hooke's Law to the case of simple longitudinal 
stress, — such as the case of a bar stretched by simple longitudinal 
pull, — we may measure the state of strain by the change of length 
per unit of original length which the bar undergoes when stressed. 
Let the original length be I and let the whole change of length be 
hi when a stress is applied whose intensity p is within the elastic 

limit. Then the strain is measured by y , and this by Hooke's 

Law is proportional to the intensity of the pull p. 

Ihus -j oc jj, 

which may be written 

Sl = p 
I ~ E' 

where E is an appropriate constant depending on the particular 
material dealt with. The same value of E applies in push as in 
pull, these two stresses being essentially of the same kind and 
only differing in sign. The longitudinal extension per unit ol 

length of -j may be conveniently expressed by a single symbol c. 

_M_p 
e ~ l~ E' 

The constant E may be defined as the ratio of the intensity "t 
stress to the longitudinal strain: — 



12 STRESS AND STRAIN. 

17. Young's Modulus or the Stretch Modulus. This 
constant E is called Young's modulus, the modulus of longitudinal 
extensibility, or more briefly the Stretch Modulus. Its value, 
which is expressed in the same units as are used to express in- 
tensity of stress, may be measured directly by exposing a long- 
sample of the material to longitudinal pull and noting the exten- 
sion, or indirectly by measuring the flexure of a loaded beam of 
the material, or by experiments on the frequency of vibrations. 
Practical methods of making such measurements will be described 
in a later chapter. It is frequently spoken of by engineers simply 
as the modulus of elasticity, but this name is too general, as there 
are other moduluses which relate to other modes of stress. 

In iron and steel the value of Young's Modulus is about 13000 
tons per square inch ; in other words, a stress of 1 ton per square 

inch produces an extension which is ^ Qnnn of the original length. 

This will serve to illustrate the important fact that the elastic 
strains which occur in engineering structures are very small 
quantities. A strain amounting to as much as one part in a 
thousand would be exceptionally large. To produce this would 
require in steel a stress of 13 tons per square inch, and the stresses 
which are permitted in structures have rarely more than about 
half this intensity. 

18. Ratio of Lateral Contraction to Longitudinal 
Extension in Simple Pull. In a stress of simple pull or push 
the width and thickness of the piece change by amounts which 
bear (for strain within the elastic limit) a definite proportion 
to the longitudinal strain. If the stretch per unit of length in 
an elastic strain is e the transverse contraction per unit of the 
width is 

(7 <lE ' 

where a is a coefficient to be determined by experiment. Its 
value in metals is generally between 3 and 4. The ratio of lateral 

contraction to longitudinal extension in elastic strains, or - , is 

<j ' 

often called Poisson's Ratio. 

When a pull is applied which exceeds the elastic limit, lateral 
contraction still accompanies the longitudinal extension but its 



STRESS AND STRAIX, 



13 



proportion is no longer the same as that which holds for elastic 
strain. All that has been said here about the lateral contraction 
produced by stress of a simple pull applies also to the lateral 
expansion produced by a stress of simple push. 

19. Strain produced by Shearing Stress. Modulus of 
Rigidity or Shear Modulus. When the state of stress is one of 
simple shear (§11) the material is distorted so that an element 
originally cubical becomes lengthened in one diagonal direction, 
and shortened to an equal extent in the other, its sides remaining 
parallel. There is no change of volume in this distortion. 

The square side ABCD (fig. 8) takes the form indicated by 
the dotted lines, its angles changing by a 



IT 



small quantity <£ to the values — + cf> and 

— — <f>. This change of angle expressed in 

circular measure serves as a measure of the 
strain : it is called the Angle of Shear. Since 
Hooke's Law holds good (within the elastic 
limit) for shear as well as for other strains, 
<f> is proportional to the intensity q of the 
shearing stress. It may therefore be written 



Fig. 8. 



*- 



9. 
C 



where C is a constant for the particular material, expressing its 
elastic resistance to shearing strain. G is called the Modulus 
of Rigidity. It is stated numerically in the same units as 
are used to specify the stress. The value of C is most often 
determined by experiments on torsion, and it is generally found 
to be about two-fifths that of Youngs modulus E. G may be 

defined as the ratio of shearing stress to shearing strain, or . 

20. Modulus of Cubic Compressibility or Bulk Modulus. 
When three simple stresses of equal intensity p and o{ the same 
sign (all pulls or all pushes) are applied in three directions, 
the material (provided it be isotropic, ili.it is to say, provided 
its properties are the same in all directions) suffers change 



14 STRESS AND STRAIN. 

of volume only, without distortion of form. If the volume 

BV 

is V and the change of volume is BV, the fraction -~ measures 

BV 

the strain. The ratio of the stress p to the strain -~ for elastic 

V 

changes of bulk is called the modulus of cubic compressibility or 

bulk modulus and may be denoted by K. 

BV_p_ 
V~ K' 

In this strain the linear dimensions of the body all change equally, 
and (assuming the cubic strain to be small) the amount of the 
linear strain in any direction is one-third of the cubic strain or 

P 
3K' 

The modulus K may be directly measured by observing the 
contraction of volume which a body undergoes when immersed in 
a liquid to which pressure is applied, but its value is more usually 
inferred from a knowledge of the other elastic constants. 

21. Relation between the Moduluses of Elasticity. The 

four elastic constants which have now been defined, namely, E, C, 
K, and a, are related in this way that if any two of them are 
known by experiment the other two may be calculated. In other 
words, an equation may be formed connecting any three of these 
constants with one another. Two of the constants are sufficient to 
specify the elastic properties of the material. The relations which 
exist between the various elastic constants will be discussed in the 
next chapter. 

22. Work done in producing an Elastic Strain : 
Resilience. When a material which follows Hooke's Law is 
strained the stress must increase in proportion to the strain, and 
the mean value of the stress is half the final value. The work 
done is measured by the product of the strain into the mean 
value of the stress. Consider a tie-rod or other piece subjected to 
simple pull, of intensity p. The strain e, per unit of length, is 

^ . This extension is produced, in each filament of unit sectional 

area, by the application of a force p and the mean force during the 



STRESS AND STRAIN. 15 

extension is \p. Hence the work done, per unit length of such a 
filament, or in other words, per unit of volume of the material, is 

p> 

2E' 

Since the material is, by supposition, elastic, this expression also 
measures the energy stored in the piece in consequence of the 
strain and capable of being restored when the strain is relaxed. 
It is called the resilience of the piece. 

In the same way — - f measures the work stored, per unit of 

volume, in a material subjected to elastic shear by a shearing 
stress of intensity q. This may readily be seen by considering 
the distortion of a block like that of Fig. 8. The forces ou the 
opposite sides of the block form a couple, and the work done is 
half the product of the angle of shear into the moment of the 
couple. By taking a block, each of whose sides has unit leugth, 
the above expression for the resilience in a shear is at once 
obtained. 



CHAPTER II. 



RELATIONS BETWEEN THE ELASTIC CONSTANTS. 

23. Relation between E, C, and K. To find a relation 
between Young's Modulus E, the modulus of rigidity C\ and the 
modulus of cubic compressibility K, the following artifice is con- 
venient. Suppose a stress of simple pull to be applied to a body 
and consider a cubical element two of whose faces are perpendicular 
to the direction A B of the applied stress. The applied stress p 
which acts on the top and bottom faces of the cube may be broken 
up into three parts each equal to ^p. Further, it will not affect 
the actual state of stress if we suppose a pull stress of intensity 
^p and also a push stress of the same intensity to be applied to 
each of the other two pairs of faces. We thus obtain the group of 
stresses indicated by arrows in the figure (fig. 9). Now the push 
of ^p on the front and back faces ah and bg together with one 
of the pulls of lp on the top and bottom makes up a state of 
simple shear the intensity of which is ±p. Again, the push of ±p 
on the two sides (if and dg together with another of the pulls of 
^p on the top and bottom makes up another simple shear, at right 
angles to the first, and also of intensity ^p. What is left is a pull 
of J-_p on every one of the six faces, that is to say, a stress producing 
cubic dilatation. 

Thus a simple pull of intensity p is found to be equivalent 
to two shears each of \p, the directions of which are at right 
angles to each other and are inclined at 45° to the axis of the 
pull, together with a cubic dilating stress which also has the 
intensity \p. 

Next we have to find the total change of length which the 



RELATIONS BETWEEN THE ELASTIC CONSTANTS. 



17 



block undergoes along the axis AB in consequence of the stresses 
due to these two shears and to the cubic dilating stress. 




Bv 

V 

Fig. 9. 



ha . 



Each of the two shears causes an extension - - in the direction 

a 

AB, where a is the diagonal of an element such as that sketched 

in Fig. 10 and 8a is the extension of 

the diagonal caused by the shear. But 

— = y (see the figure) and is there- 
fore equal to \<f>. 

Further, </> = ~ , since J p is the 

intensity of the shearing stress. 

Hence each of the shearing stresses 
extends the piece in the direction AB 
by the amount (per unit of length) 

1 P 

o • 




Fig. 10. 



B>, B. BI. 



18 RELATIONS BETWEEN THE ELASTIC CONSTANTS. 

Again, the cubic dilating stress ±p extends every dimension by 
the amount (per unit of length) 

K' 

Thus the whole strain in the direction AB is equal to 

9 iP ,iP _ ( 1 1 \ 

due to the two shears, and the cubic dilating stress which we have 
seen to be equivalent to the simple pall p. 

But the strain along AB due to a simple pull p is directly 

expressed as 

P 
E' 

Hence 4=^ = p ( ^ + tt~) , from which 

F _ 9KC 
SK+C 

24. Relation of a to the moduluses C and K. Consider 
next the change in transverse dimensions due to the shears and 
the cubic dilatation into which we have analyzed the strain. Each 
transverse dimension is affected by one of the two shears. The 

shear contracts it to the extent ^ and the cubic dilatation enlarges 

i P 

it to the extent ~^ . 

Hence the resultant lateral contraction, perpendicular to the 
direction A B of the applied stress p, is 

JL_P_ 
6(7 9K' 

and a the ratio of the longitudinal extension to the lateral contrac- 
tion is given by the equation 



SC + 9K QK+2C 



a = 



1^ 1_ SK-2C 
60 9K 

It follows from this that o is always greater than 2, since C is 
necessarily positive. It will be only slightly greater than 2 in 
substances which have K great in comparison with 0. An instance 
in point is furnished by india-rubber, where the elastic resistance 



RELATIONS BETWEEN THE ELASTIC CONSTANTS. 19 

to change of volume is great and the resistance to shearing is 
small. India-rubber is distorted with ease but compressed (cubi- 
cally) with great difficulty, and consequently the value of a in 
it is not far short of 2. 

The fractional expansion of volume which a body undergoes in 
the elastic strain caused by a simple pull along one axis is equal to 

2 

the longitudinal strain multiplied by 1 , since for every unit by 

which the length of a cubical element increases the breadth and 
thickness each diminish by - . No change of volume would result 
if a were equal to 2. 

25. Other expressions of relation between the Elastic 
Constants. The equations given above are sufficient to establish 
a connection between any three of the four elastic constants. It 
may, however, be useful to indicate other ways in which these 
relations may be found and other forms in which they may be 
expressed. 

Consider for example a simple shear of intensity p. The 
linear extension and contraction respectively in two directions 

inclined at 45° to the plane of shear are ^ or ^. We may 

regard the shear as made up of two principal stresses, one of push 
and one of pull, along these two directions, each having the same 
intensity as the given shearing stress. The strain in either direc- 
tion is 

p £_ 

E ' o-i," 

the first term representing the direct effect of the one principal 
stress and the second term representing the lateral effect of the 
other. 

Hence 2C~~ E* aE y 

or E = 2Ch+^ 

and C = 



2(o-+ 1)' 

This equation may of* course bo obtained also 1>\ eliminating 
K in the results of §§ 23 and 24. 

2 2 



20 RELATIONS BETWEEN THE ELASTIC CONSTANTS. 

It may be concluded from this that is always less than 

— . Further, since a is not less than 2, the factor (1 + - ) cannot 
^ V a- J 

exceed f. Hence E has in all cases a value lying between 2C 
and 30. In metals <t is usually about 3 J, which makes E about 
2-60. 

Again, consider a piece under "fluid" stress, namely with 
three equal principal stresses all of the same sign. The linear 
strain in any direction is made up of one direct and two lateral 
strains and has the value 

E aE crE E\ <r J' 
Hence the volume strain, being three times the linear strain, is 

But the volume strain may also be expressed as ^ , whence 

A-Afi-*) 

K E\ <rJ' 
or E = SK (l - 1 

and K = 



3 (a - 2) ■ 

Combining this with the equation given above connecting C and 
E with o-, and eliminating E, we have, 

2(SK+C) 



a = 



SK - 20 
or, eliminating a, 

3K+C 

These are identical with the results obtained more directly by 
another method in §§ 23 and 24. 

26. Isotropic material : Equations connecting Stresses 
and Strains. Suppose an isotropic material to be in a state of 
stress of the most general kind, and let the three principal stresses 
be^,^, said p z , the axes of reference being chosen so that they 
coincide with the principal axes. Let the strains along these axes 
be e x , e y and e z respectively. Then e x is made up partly of the 



RELATIONS BETWEEN THE ELASTIC CONSTANTS. 21 

direct strain which p x produces in its own direction, and partly of 
the lateral strains produced by p y and p z . The direct strain due 

to p x is ¥§ and the lateral strains are — ^ an d — if! • 
Jcj ah ah 

Thus e =P?- P» + P* 

or Ee x =p x -Vz±&. 

Similarly Ee y =p y - 2*±£* t 

and Ee z = p z -^- Py . 

a 

We proceed to apply these general equations to particular cases. 

27. State of Simple Shear. The state of stress is one of 
simple shear when p y = — p x and p z = 0. In that case the 
equations become 

Ee x =p x +^ = p z ML + -J, 
Ee y = -p x - P - = - Vx ( 1 +-) , 



Ee z = 0. 

We have seen (§ 23) that the strain along each axis of principal 
stress is equal to \(f> where d> is the angle of shear. 

2px i 1 + l) 
Hence c/> = 2e x = /T , 

and since the intensity of the shearing stress is numerically equal 
to p x or p y , this leads, as before, to the equation 

E=2C(\ + - 

28. Volume Strain. By adding the three general equations 

we have 

E (e x + e y + e z ) = {p x + p y + p 9 ) (l - -J , 
or E-y=(P»+Pv + P*)[} -~ 



22 RELATIONS BETWEEN THE ELASTIC CONSTANTS. 

In the particular case when Px=p y =p z (the case which occurs 
in a fluid) 

8V „ / 2^ 



which gives the relation already found in § 25 

(7 

29. Simple Strain along one axis. The student will 
notice that a simple strain, in the sense of a strain along one 
axis only, is not produced by the application of a simple stress. 
To restrict the strain to e x , making e y and e z vanish, we have to 
apply certain stresses p y and p z as well as a stress p x . The 
equations become in that case 



0=p 2 - 



Px+Py 



Px 



Hence p y = p z — -. > showing that a simple contraction without 

lateral expansion is produced when a direct compressive stress is 
associated with two equal lateral stresses, also of compression, 
each of which is less than the direct stress in the ratio 1 : a — 1. 

The relation which then holds between the strain e x and the 
stress p x is given by the equation 

j? /-, 2__\ . (o--2)(<r + l) 

Ee x =p x [l t 7T = p 3 



Hence 



o-(o--l)/ iX o-(c--l) 
^ ^cr(o--l) 



e x ((7-2)(c7 + l) 5 

and this constant may be called the modulus of elasticity for the 
special mode of stress here assumed. Under this mode of stress 
the strain e x is less than that which would be produced by p x> 
acting alone, in the ratio of a 2 — a — 2 to a' 2 — a. 

30. Lateral Strain prevented in one direction. A case 
presenting more practical interest is found when a piece, pulled or 
pushed along one axis (OX), is left free to contract or expand 



KELATIONS BETWEEN THE ELASTIC CONSTANTS. 23 

along one lateral axis (OY) but is prevented from changing its 
dimension along the other lateral axis (OZ). In that case e 2 and 
p y are zero and we have 

Ee x =p x - 1 -, 
a 

Be,—***, 

a 
Hence the required lateral strain p z is of the same sign as p x 

TO 

and is equal to — , and 
c 

Ee x =p x (1 



Thus the special modulus of elasticity for this mode of stress is 



-2 
-I' 



and the strain e x is less than p x acting alone would produce, in 
the ratio of cr 2 — 1 to a-. 

31. Numerical Example. The elastic constants which are 
most usually measured are Young's modulus E and the modulus 
of rigidity C. In a later chapter experiments will be described 
by which such measurements are made. With steel, the modulus 
E is about 13000 tons per square inch and C is about 5000 tons 
per square inch. Taking these values we may determine corre- 
sponding values of the other constants, thus — 

EC 
■^ = 7ui — 7Tny = 10830 tons per square inch, 

2C 
E-2C 

The modulus for a strain in which lateral contraction or ex- 
pansion is entirely prevented, or ^ ~T\> ls 17500 tons per 

(a — z) (cr-f 1 ) 

square inch, and the modulus for a strain in which lateral 

contraction is free to take place in one direction but prevented 

Eg' 1 
in the other, or — — -, is 1421)0 tons per square inch. 

<T J — 1 



a - -& an ~ &h 



CHAPTER III. 

ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 

32. Strain carried beyond the Limit of Elasticity. All 

that has been said about the elastic constants refers only to the 
strains which occur when the stresses are so small as to fall within 
the limits of elasticity. Within these limits we may without 
serious inaccuracy take the strain as being proportional to the 
stress and as disappearing when the stress is removed. Strictly 
speaking, absolute proportionality of strain to stress is never 
found, and probably there is no stress, however small, that does 
not produce some permanent effect. There is always some slight 
hysteresis or lagging in the relation of strain to stress, which 
shows itself for example when a tie-bar is alternately loaded and 
unloaded, the length under any intermediate amount of load 
being a very little greater during unloading than during loading. 
But in general this imperfection in elasticity is so slight that it 
may safely be disregarded when we are dealing with the strains 
caused by comparatively small amounts of stress, and up to a 
certain limit, which is in general pretty well defined, Hooke's 
Law may be taken as substantially accurate. 

When that limit is reached a change takes place in the 
relation of strain to stress which exhibits itself in two ways. As 
the loading proceeds the increments of strain, for equal increments 
of stress, become greater, and further it is observed that the 
amount of strain due to any given load depends to some extent 
on the time during which the load acts. When a load exceeding 
the elastic limit is applied the strain which occurs at once is 
followed by a continued "creeping" or supplementary deformation 
which is very noticeable during the first few minutes and may go 
on, though at a diminished rate, for a much longer time. 



ULTIMATE STRENGTH AXD NON-ELASTIC STRAIN. 25 

We shall call a piece overstrained when the stress exceeds the 
elastic limit. The overstrained piece is more or less plastic. A 
stress which produces overstraining requires a long time (possibly 
an indefinitely long time) to produce its full effect. As the load 
is further increased this plasticity becomes in general more and 
more marked. The material in some cases flows under the applied 
stress like a viscous liquid and time is the main factor in de- 
termining the amount of the strain. The behaviour of iron, mild 
steel, and most other metals when tested for tensile strength, 
exemplifies this. When the load exceeds the elastic limit a 
certain amount of " creeping " or continued extension is observed 
whenever the process of loading is temporarily suspended. And 
a stage is reached at which, without further increase of load, the 
piece continues to draw out until it breaks. 

33. Ultimate Strength. The load which suffices to cause 
rupture measures the ultimate strength of the piece. In 
reckoning the ultimate strength of a material in tons or pounds 
per square inch the practice of engineers is to take, not the 
actual intensity of stress at the time when the piece breaks, but 
the value which this intensity would have reached had the 
original area of section remained unchanged. In other words, 
the ultimate strength is reckoned as the breaking load per square 
inch of the original area of section, not per square inch of the 
area which the section has when the piece breaks. Thus if a bar 
whose original cross-section is 2 square inches be broken by 
applying a total pull of 60 tons, uniformly distributed over the 
section, the ultimate tensile strength of the material is said to be 
30 tons per square inch, although the actual intensity of stress in 
the last stages of the test may have been much greater than this 
in consequence of the contraction which the section undergoes 
before the piece breaks, especially in the neighbourhood where 
the break is to occur. 

The reason for this usage is that engineers wish in all cas< - 
to know what total load will break a piece, in order thai they may 
arrange to prevent the actual load from being more than a safe 
fraction of that. Suppose for instance that a tie-rod is t<» be 
designed to bear safely a pull of 12 tons, and that the working 
load is to be only one-fifth <>f the Load which would break the 

rod. It must in that case have such an area of -'•(•lion as would 



26 ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 

be broken by a load of 5 x 12 or 60 tons. Hence if the material 
has an ultimate strength of 30 tons per square inch, the proper 
area of section is at once seen to be §£ or 2 square inches. Or, to 
put the same thing in a slightly different way, the working load 
is to be one-fifth of 30, or 6 tons per square inch, and the section 
is accordingly -i£ or 2 square inches. Such a calculation proceeds 
on the basis of the ultimate strength as defined by reference to 
the original area of section of a test-piece, and has nothing to do 
with the changes of section which occur during the process of 
testing. It may be added that the working loads on the parts of 
engineering structures are, or ought to be, in all cases within the 
limits of elasticity, and within these limits the change of cross- 
section caused by the elastic strains is so small that it may be 
neglected in calculating the intensity of stress. 

Ultimate tensile strength and ultimate shearing strength are 
well defined, since in the corresponding modes of stress, namely 
simple pull and simple shear, a distinct fracture is observed when 
the stress is sufficiently increased. Under compression, on the 
other hand, some materials yield so continuously that their ultimate 
strength to resist compression can scarcely be specified : it would 
for instance be difficult to assign any value to the compressive 
strength of such a substance as lead, for a test-piece under 
compression would flatten out almost without limit. Some 
materials, notably brick and stone as well as the more brittle 
metals, show so distinct a fracture by crushing that their com- 
pressive strength may be specified with fair precision. 

Some of the materials used in engineering are so far from 
being isotropic that their strength is widely different for stresses 
in different directions. The tensile strength of timber, fur 
example, is immensely greater when pull is applied along the 
fibre than when pull is applied across the fibre, and a similar 
difference exists in regard to the shearing strength. In wrought- 
iron the process of rolling developes something of a fibrous 
structure, partly in consequence of the presence of streaks of 
slag which become drawn out into long lines as the bar or plate 
is rolled. The tensile strength of a rolled iron plate is accord- 
ingly found to be considerably greater in the direction of rolling 
than across the plate. Steel plates, being rolled from a nearly 
homogeneous ingot, are more nearly isotropic, but even in them 
some difference of the same kind is observed. 



ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 27 

34. Factor of Safety. In applying a knowledge of the 
ultimate strength of materials to determine the proper sizes of 
parts in an engineering structure the designer has to select a 
number which will express the ratio of the ultimate strength to 
the stress which is to be allowed. In other words, the working 
stress, as we may for brevity call the stress which will occur in the 
loaded structure, has to be a certain fraction of the ultimate 
strength, and what this fraction should be is a matter for the 
judgment of the engineer. The ratio 

ultimate strength 
working stress 

is called the factor of safety. 

The choice of a factor of safety depends on many considerations, 
such as the probable accuracy of the estimated loads and also that 
of the theory on which the calculation of the working stress has 
been based ; the uniformity of the material dealt with, and the 
extent to which its strength may be expected to conform to the 
assumed value or to the values determined by experiments on 
samples ; the possible effects of bad workmanship in causing a 
deviation from the specified dimensions when the structure is 
actually built; the degree to which the materials may be expected 
to deteriorate in time or by exposure to variations of temperature. 
Another important consideration in the choice of the factor of 
safety is the variability or uniformity of the load : for reasons 
which will appear presently a larger factor is properly chosen 
when the load is subject to repeated changes. The factor of 
safety also serves to provide for the incidental shocks which 
may occur in consequence of sudden variations in the load. 
Such shocks cause supplementary stresses which can scarcely 
be made subjects of calculation. 

The factor of safety is rarely less than 3, it is very commonly 
4 or 5, and it is sometimes as much (in machines) as 10 or L2. 
A Board of Trade rule permits the working stress in bridges and 
other structures of wrought-iron to be 5 tons per square inch. 
As the tensile strength of the material is in this case about 
20 tons per square inch, the rule corresponds t<> a factor of safety 
of about 4. A committee of engineers reporting to the British 
Association in ]cStt7* recommends that in small bridges, where 

* Rep. Brit. 4moc, L887, ]>• 488, 



28 ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 

the permanent load due to the weight of the structure is small 
compared with the variable load due to traffic, the working stress 
should not exceed 4 tons per square inch, but that in the 
case of bridges or other structures of such magnitude that the 
dead weight is more than twice the moving load the working 
stress may safely be increased to nearly 6 tons per square inch. 
These numbers correspond to factors of safety of about 5 and 3-J- 
respectively. In the design of the Forth Bridge, where steel was 
employed having a tensile strength of from 30 to 33 tons per 
square inch, the working stress was allowed to reach 7J tons per 
square inch, but in members liable to alternate compression and 
extension it was restricted to 5 tons per square inch. 

The ratioDal use of a factor of safety in determining the 
dimensions of the several parts of a structure results in not only 
making all parts sufficiently strong, but in preventing waste of 
material locally by making the margin of strength equal for all 
parts. 

35. Variation of the Ultimate Strength under different 
modes of loading. In specifications of ultimate strength it is 
generally assumed that the load is made to increase continuously 
and at a fairly rapid rate until the piece breaks : it is supposed to 
be applied as it would be applied in ordinary testing. But by 
following special modes of loading it is possible either to increase 
or to diminish the ultimate strength very considerably. Instances 
of this will be detailed later. By adding the load in a series of 
steps with long pauses betw r een we may cause the piece to bear 
much more than would suffice to break it if applied in the usual 
way. On the other hand, if a load be applied and removed many 
times it will suffice to break the piece even though its amount is 
much less than would be needed to cause rupture in a single 
application. A much smaller stress still will cause rupture if 
it alternates between compression and extension. Hence in a 
structure which has to bear " live " or variable load the per- 
missible intensity of working stress is less than in a structure 
which bears only " dead " or constant load. 

36. Advantage of Plasticity. From an engineering point 
of view the structural merit of a material, especially when live 
loads and possible shocks have to be borne, depends not only on 



ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 29 

the ultimate strength but also on the extent to which the material 
will bear deformation without rupture. Other things being equal, 
a material which in the later stages of the test exhibits much 
plasticity, by drawing out much under pull and narrowing its 
section before it breaks, is to be preferred to one which breaks 
off short. Accordingly the ultimate elongation and the contraction 
of area are often specified as well as the ultimate strength. The 
same characteristic is often tested in other ways, such as by 
bending and unbending bars in a circle of specified radius, or 
by examining the effect of repeated blows. This is sometimes 
done by supporting a piece of the material on a beam and 
causing a weight to fall on the middle of it from a given 
height. 

« 

37. Ordinary Tensile Tests. The most usual test however 
is made by applying a direct pull and gradually increasing it until 
the specimen breaks. When the samples to be tested are small 
wires the stress may be applied directly by hanging up the wire 
and applying weight, but when larger sections are to be dealt 
with some form of testing machine is needed to facilitate the 
application and measurement of the load, and to allow the con- 
siderable amount of work to be done which is expended in drawing- 
out the piece beyond the limit of elastic strain. 

As the test proceeds the extension is at first so small that it 
can be measured only by a microscopic or other refined apparatus 
called an extensometer. Presently the elastic limit is passed : the 
increments of strain then become greater and the phenomenon of 
creeping begins to be observed. In plastic metals such as wr< night- 
iron and steel a further change happens when the load is increased 
to a somewhat higher value than the elastic limit. A point called 
(by Professor Kennedy) the yield point is reached at which the 
specimen draws out suddenly, the sudden increase of extension 
being generally greater than the whole amount of the extension 
caused by smaller loads. At this stage, and during the remainder 
of the test, the extension is usually so great that a pair of compasses 
and a foot-rule serve to measure it. After the yield point is 
passed, the piece continues to extend more or less irregularly 
under augmented loads until rupture is about to take place, At 
that stage there is a supplementary local yielding the portion 
in the neighbourhood of the place where fracture is to orciii- 



30 



ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 



draws out much more rapidly than the rest of the bar, and the 
section there becomes attenuated. 

38. Diagrams of Extension and Load in Tensile Tests. 

The chief results of a tensile test are conveniently exhibited by 
drawing a diagram to show the relation of the extension to the 
load (reckoned per square inch of the original area of section of 
the specimen). Typical diagrams for wrought-iron, mild steel, 
and comparatively hard steel are given in fig. 11, the data for 
which are taken from tests by the late Mr David Kirkaldy*. Up 



45 



40 



q35 



30 



25 



z20 

O 

h 



- 15 



10 





4 


















< 


f 


























MIL, 


)0£SSi 


■MER S 


TEEL 






































(LOKg 


tforyil 


2) 












^ 


*jj*- 











































































6 8 10 12 14 

EXTENSION, PER CENT 



16 



18 



20 



Fig. 11. 

to the elastic limit the extension per ton of load is much the 
same in all three materials, so that the early portions of the three 
curves are indistinguishable. In each case there is a visible defect 
of elasticity some way before the yield point is reached, and then 
a well-marked yield point, especially in the softer metals, at which 
extension goes on for a time through a considerable distance with- 
out increase of load. After this the extension becomes less rapid 

* Experiments on the Mechanical Properties of Steel by a Committee of Engi- 
neers ; London, 1868 and 1870. 



ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 



31 



until the final yielding occurs just before rupture. In the last 
stages the flow of the metal continues even if the load be some- 
what reduced, and it is therefore possible to make the end of the 
curve bend back by taking off part of the load as the end of the 
test is approached. 

By way of contrast with the diagrams of extension and load in 
plastic metals, shown in fig. 11, reference should be made to fig. 12, 









-. 10 
o> 

CO 
C£ 
LU 

to 












§ 5 

co~ 

CO 

in 

1— 

CO 


! / 




•4 
CO 


3 
1PRESSI0N 


■2 

PERCENT 


1 / 

/ ' 
/ / 


"o 

EXTENSA 


1 

.PERCENT 








1 

/ 


QUARE INCH 






vY 




/ 
/ 
/ 


o 
TONS PER S 




S 






/ 


15 - ± 
a 
o 

20 





Fig. 12. 

which shows how cast-iron behaves under compression as well as 
under tension. The figure is taken from one of Hodgkinson's 
experiments*. 

The extension was measured on a rod 50 feet long ; the 
compression was also measured on a long rod, which was pre- 
vented from buckling by being supported in a trough with 
partitions. The full line gives the strain produced by loading; 
it is continuous through the origin, showing that Young's modulus 
is the same for pull and push — a result which is also found to 
hold good in other materials. The broken line shows the set 
produced by each load. Hodgkinson found that in cast-iron some 
set could be detected after even the smallest- loads had been 
applied. This is probably due to the existence of initial internal 
stress in the metal, produced by unequally rapid cooling in 

* Report of the Commissioners on ///<■ Application of Iron to Railway Structures, 

1840. 



32 ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 

different portions of the cast bar. A second loading of the same 
piece showed a much closer approach to perfect elasticity. The 
elastic limit is, at the best, ill defined ; but by the time the 
ultimate load is reached the set has become a more considerable 
part of the whole strain. The pull curves in the diagram extend 
to the point of rupture ; the compression curves are drawn onlv 
up to a stage at which the bar buckled between the partitions so 
much as to affect the results. 

39. Autographic Diagrams of Extension and Load. 

Testing machines are frequently fitted with recording appliances 
for automatically drawing diagrams showing the relation of the 
extension to the load. When the load is measured by a weight 
travelling on a steelyard, the diagram may be drawn by connecting 
the weight with a drum by means of a wire or cord, so that the 
drum is made to revolve through angles proportional to the travel 
of the weight. At the same time another cord, fastened to a clip 
near one end of the specimen, and passing over a pulley near the 
other end, draws a pencil through distances proportional to the 
strain, and so traces a diagram of stress and strain on a sheet of 
paper stretched round the drum. 

Apparatus of this kind is serviceable in showing the behaviour 
of plastic materials after the elastic limit has been passed. Effects 
of viscosity can be traced by noticing the changes in the form of 
the curve when pauses are made during the application of the 
load. The full strain corresponding to a given load is reached 
only after a perceptible time, probably a long time. If the load 
be increased to a value exceeding the elastic limit, and then kept 
constant, the metal will be seen to draw out (if the stress be one 
of pull), at first rapidly and then more slowly. When the applied 
load is considerably less than the ultimate strength of the piece 
(as tested in the ordinary way by steady increment of load), it 
appears that this process of slow extension comes at last to an end. 
On the other hand, when the applied load is nearly equal to the 
ultimate strength, the flow of the metal continues until rupture 
occurs. Then, as in the former case, extension goes on at first 
quickly, then slowly, but, finally, instead of approaching an asymp- 
totic limit, it quickens again as the piece approaches rupture. 
The same phenomena are observed in the bending of timber and 
other materials when in the form of beams. If, instead of being 



ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 



33 



subjected to a constant load, a test-piece is set in a constant 
condition of strain, it is found that the stress required to maintain 
this constant strain gradually decreases. 

The gradual flow which goes on under constant stress — 
approaching a limit if the stress is moderate in amount, and 
continuing up to fracture if the stress is sufficiently great — will 
still go on at a diminished rate if the amount of stress be reduced. 
Thus, in the testing of soft iron or mild steel by a machine in 
which the stress is applied by hydraulic power, a stage is reached 
soon after the limit of elasticity is passed at which the metal 
begins to flow with great rapidity. The pumps often do not keep 
pace with this, and the result is that, if the lever is to be kept 
floating, the weight on it must be run back. Under this reduced 
stress the flow continues, more slowly than before, until presently 
the pumps recover their lost ground and the increase of stress is 
resumed. Again, near the point 
of fracture, the flow again be- 
comes specially rapid ; the 
weight on the lever has again 
to be run back, and the speci- 
men finally breaks under a 
diminished load. These features 
are well shown by fig. 13, which 
is copied from the autographic 
diagram of a test of mild steel*. 

40. Hardening effect of 
rest after overstraining. But 

it is not only through what we 
may call the viscosity of ma- 
terials that the time rate of 

loading affects their behaviour under test. In iron and steel, and 
probably in some other metals, time has another effect of a very 
remarkable kind. Let the test be carried to any point a (tig 14) 
past the original limit of elasticity. Let the load then be removed : 



EXTENSION 



Fig. 13. 



* The increase of strain without increase of stress, which g06fl Oil without limit 
when a test-piece under tension approaches rupture, is a Bpeoial oase oi' the general 
phenomenon of " flow of solids," which baa been exhibited, ohieflj for oompn 
stresses, in a series of beautiful experiments by Treeoa {Mimoire* tur VEcoulement 
det Corps Bolides, also l'nir. Inst. Mech. Etig, L867 and 1878. 

B. B. m. 3 



34 



ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 



during the first stages of this removal the material continues to 
stretch slightly, as has been explained above. Let the load then 
be at once replaced and loading continued. It will then be found 
that there is a new yield-point b at or near the value of the load 
formerly reached ; up to this point there is but little strain. The 
full line be in fig. 14 shows the subsequent behaviour of the piece. 
But now let the experiment be repeated on another sample, with 



o 

z 

bJ 

5 20 

(A 



0. 15 







l~e 


c 




/a 


% 





































O 5 IO 15 20 

EXTENSION, PER CENT 

Fig. 14. 



25 



?20 

D 
O" 
W 



w 



15 



10 







c...d 






■H 



























O 5 10 15 

EXTENSION, PER CENT 



Fig. 14 a. 



this difference, that an interval of time, of a few hours or more, is 
allowed to elapse after the load is removed and before it is replaced. 
It will then be found that a process of hardening has been going 
on during this interval of rest ; for, when the loading is continued, 
the new yield-point appears, not at b as formerly, but at a higher 
load d. Other evidence that a change has taken place is afforded 
by the fact that the ultimate extension is reduced and the ultimate 
strength is increased (e, fig. 14). 

A similar and even more marked hardening occurs when a load 
(exceeding the original elastic limit), instead of being removed 
and replaced, is kept on for a sufficient length of time without 
change. When loading is resumed a new yield-point is found only 
after a considerable addition has been made to the load. The 
result is, as in the former case, to give greater ultimate strength 
and less ultimate elongation. Fig. 14 a exhibits two experiments 



ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 35 

of this kind, made with annealed iron wire. A load of 23J tons 
per square inch was reached in both cases ; ah shows the result of 
continuing to load after an interval of five minutes, and acd after 
an interval of 45J hours, the stress of 23^ tons being maintained 
during the interval in both cases*. 

41. Breakdown and Recovery of Elasticity after 
overstraining. Further, a piece of iron or steel which has been 
overstrained, so that permanent set has been produced, is found to 
be very imperfectly elastic even with respect to small amounts of 
load, when, after being removed, the load begins to be reapplied. 
Hooke's law does not then hold even for loads much below the usual 
elastic limit of the material. When a small load is applied the 
immediate strain is seen to be followed by slow creeping, and if 
the load be removed the strain which it caused does not imme- 
diately disappear, but there is a slow creeping back. This is the 
state of things just after the molecular structure of the piece has 
been disturbed by overstraining. But if, after a considerable 
interval of time (such as a few days), the overstrained piece is 
tested again, a partial recovery of elasticity is found to have taken 
place, and this recovery becomes more and more complete as 
time goes on-f*. The following experiment will serve to show 
the character of this action, as regards both the immediate effect 
of overstraining in depriving the material of its usual elasticity 
and also the subsequent recovery of elasticity with lapse of time. 
The readings quoted were taken with an extensometer by which 
the extensions of a nine-inch length in the middle of the specimen 
were read to the nearest fifty-thousandth of an inch. The zero- 
reading of the extensometer was 200. The specimen was a turned 
rod of semi-mild steel, with a diameter of 0*705 inch (section 
0*390 sq. inch). In the initial test of the piece, before over- 
straining, it was found that Hooke's law held with great accuracy 
up to a load of 10 tons (corresponding to a stress of 25*6 tons per 
square inch) and that the average extension per ton of load up to 

* The experiments of figs. 14 and 14 a are taken from a paper by tin' author in 
Proc. Hoy. Soc. 1880, "On Certain Effects of Stress on Soft Iron Wins," where 
further experiments bearing on the same point will be found. 

t For an investigation of this effect of overstraining, sec papers by Bausohinger, 
Mitth. mis dem mech.-tech. I. ah. in MilTichen, ami by the author, /v<><-. Roy. Soc* % 
vol. 58, L895, 



36 ULTIMATE STRENGTH AND XON-ELASTIC STRAIN. 

that point was 85J of the divisions of the extensometer, corre- 
sponding to a value for the modulus E of 

1x9x50000 1Q ™. . , 

n . QQn — a ^! = looOO tons per square inch. 

\J 0'j\) X 0O9" 

The load was increased to 11 tons, when the yield-point was 
reached and a permanent extension (of 0'14 inch) took place. 
Immediately after the overstraining the load was removed, and a 
series of subsequent tests were made which are detailed in the 
Table below. In the first of these, made only a few minutes after 
overstraining had occurred, there is nothing like proportionality of 
extension to load even with loads of one or two tons, and creeping 
was observed to occur almost from the first. The later tests 
show a gradual progressive recovery of elasticity, which however 
is by no means complete even after three weeks. 

An experiment of this kind serves to emphasise the distinction 
between the yield-point and the elastic limit. If the loading of 
the overstrained piece had been continued a yield- point would 
have appeared at a load higher than 11 tons, considerably higher 
after several days of resting. But the elastic limit in this condi- 
tion, if there can be said to be any limit within which the elasticity 
is sensibly perfect, is very low — probably not higher than about 
2 tons. Immediately after overstraining the piece cannot properly 
be said to have any elastic limit, but when a period of resting 
brings about recovery, a more or less definite elastic limit re- 
appears, and rises to higher loads the more prolonged is the period 
of rest. 

In wrought-iron the recovery of elasticity after overstraining 
takes place much sooner than it does in mild or semi-mild steel. 
When in the overstrained condition, and before recovery has taken 
place, iron or steel exhibits much hysteresis in the relation of 
extension to load. Any process of loading and unloading, repeated 
until the changes become cyclic, then shows a well-marked 
difference in the length of the piece for any one amount of load 
in the two stages of the process. The curves exhibiting extension 
in relation to load form a loop, and this loop closes up as the piece 
gradually recovers its elasticity by prolonged rest. 

42. Effect of Heating in facilitating Recovery of Elas- 
ticity after overstraining. An interesting contribution to this 



ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 



37 



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38 



ULTIMATE STRENGTH AXD XOX-ELASTIC STRAIN, 



subject has been made in experiments conducted in the author's 
laboratory by Mr James Muir*, who has found that when a piece 
of iron or steel has had its elasticity broken down by overstraining 
it will make a very complete recovery if heated for a few minutes 
to a temperature such as that of boiling water. "When the over- 
strained piece has been immersed in a bath of boiling water it is 
found to have practically perfect elasticity up to a new yield-point 
which is higher than the load used in the process of overstraining. 

Figs. 15 and 15 a illustrate this by curves drawn from 
Mr Muir's observations. In the experiments of fig. 15 the 




EXTENSION (REDUCED) 
Fig. 15. Serai-mild Steel. 

material was a semi-mild steel with 04 per cent, of carbon, which 
when tested in the ordinary way showed a breaking strength of 
39 tons per square inch. In fig. 15 a the material was wrought 
iron with a breaking strength of 23 tons per square inch. The 

* Phil. Trans. Boy. Soc, 1899. 



ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 



39 



length under measurement was 8 inches in both cases, and the 
unit of extension in the diagrams is -^^ of an inch. In drawing 



30 Tons per 
sq. inch. 




A. 
B. 
C. 
D. 
E. 



Primary test. 

10 minutes after A. 

16 hours after A. 

10 minutes after C. 

After 4 minutes' 
exposure to 100° Cent. 



EXTENSION (REDUCED) 
Fig. 15 a. Common Wrought-Iron. 



40 Tons per 
sq. inch. 

35 




EXTENSION (REDUCED) 

Fig. 15 b. Hysteresis in Semi -mild Steel after over-straining. 

these diagrams the geometrical device IS used of shearing baok the 
curves uniformly by 1 unit of extension for each 4 tone pel square 



40 ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 

inch of load. The curves are drawn from separate origins to avoid 
confusion. In fig. 15 the steel bar in its first loading gave a well- 
marked elastic limit at about 22 tons, and this primary test was 
continued (far beyond the limit to which the curve A is drawn) 
until the load was 35 tons per square inch. Curve B shows the 
very imperfect elasticity which the piece exhibited immediately 
after this overstraining. In other experiments curves of loading 
and unloading were observed for this condition of the metal, and 
were found to show the characteristics of hysteresis exemplified in 
fig. 15 b, where the arrows sufficiently indicate the sequence of 
the operations. Returning to fig. 15, curve G shows the remarkably 
complete elastic recovery which results from exposure to the 
temperature of boiling water, and also the raised elastic limit 
which this treatment produces. Curve D shows a further raising 
of the elastic limit, by additional overstraining (after G) followed 
by a second bath in boiling water. 

In fig. 15 a, the first overstraining A is seen to produce the 
non-elastic state B, but a rest of 16 hours suffices to restore nearly 
perfect elasticity, and the next loading gives the curve C, with a 
raised elastic limit. This operation was carried far enough to 
overstrain the piece a second time, and curve D then shows that 
a very imperfectly elastic condition has reappeared. Finally curve 
E shows the recovery of elasticity brought about by immersion in 
boiling water. This piece was further overstrained and its elasticity 
was again restored by hot water, with the result that it finally 
bore a load of 29-J- tons per square inch before breaking. 

A remarkable experiment may be made by taking a bar of 
mild steel and stretching it in the first instance just up to the 
primitive yield-point, then heating it for a few minutes to 100° C. 
to produce elastic recovery, then stretching it again just up to its 
new yield-point, then heating again to 100° C. and so on. Each 
step raises the elastic limit, and notwithstanding its naturally 
plastic quality the bar may in this way finally be caused to break 
with a fracture resembling that of hard steel, with comparatively 
little total extension or contraction of section at the fracture, and 
under a total load much greater than that which could be applied 
in an ordinary test. 

In one of Mr Muir's experiments a bar of semi- mild steel 
showing a strength under ordinary tests of 39 tons per square inch 



ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 41 

and an extension of about 20 °/ on 8 inches, was caused to pass 
through a series of steps of this kind, by being heated to 100° C. 
just after each successive yield-point was passed. The first yield- 
point was at 27 tons, the next at 33, the next at 38, the next at 
43 J, and the last about 47. The piece was then broken, showing 
a strength of 49J tons per square inch and a total extension 
(including all the steps) of 12 per cent. 

43. Influence of time in the process of testing. We 

have seen that intervals of rest, during the process of testing, 
cause a hardening effect once the primitive elastic limit has been 
passed, and after any such interval a new yield-point appears at a 
higher load. By applying the load in a series of steps, with long- 
pauses between, the results of a test may be made to differ very 
considerably from those that are found in the ordinary process of 
continuous or nearly continuous and fairly rapid loading. The 
time during which any load (exceeding the elastic limit) is kept 
on affects the result in two somewhat antagonistic ways. It aug- 
ments extension by giving the metal leisure to flow. On the other 
hand it reduces the amount of extension which subsequent greater 
loads will cause. The stepped curve got by applying the load in 
parts with long intervals between shows (in iron and steel) a less 
total elongation and a greater ultimate strength than are found in 
the ordinary continuous process. An early illustration of this was 
given in experiments by Mr J. T. Bottomley*. Pieces of iron wire, 
annealed and of exceptionally soft quality, when loaded at the rate 
of 1 lb. in 5 minutes broke with 44£ lbs. and stretched 27 per cent, 
of their original length before breaking. Other pieces of the same 
wire loaded at the rate of 1 lb. in 24 hours broke with 47 lbs. and 
stretched less than 7 per cent. 

It does not appear that such variations in the rate of Loading 
as are liable to occur in practical tests of iron or steel have much 
influence on the extension or the strength, great as the effects of 
time are when the metal is loaded either much more slowly or 
much more quickly. In fig. 1G the results are shown of tests b\ 
the author of two similar pieces of soft iron wire, one Loaded to 
rupture in 4 minutes and the other at a rate about 5000 times 
slower. 

* Proo, Roy. Soo. t L879, p. 991, 



42 



ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 



20 



| IB 

LLl 
< 

a 

E 10 



a 5 

< 

o 





/ 


."" 








/ / 
/ ./ 

1 





































5 10 15 20 25 

EXTENSION, PER CENT 

Fig. 16. 



30 



44. Effects of Hardening through overstrain. It may 

be concluded that when a piece of iron or steel (and probably the 
remark applies to most other metals) has been overstrained in any 
way — that is to say, when it has received a permanent set by the 
application of stress exceeding its limit of elasticity — it is hardened, 
in the sense of being rendered less capable of plastic deformation. 
Further, after any such overstrain the physical properties of the 
material go on changing for days or even months — the change 
being in the direction of greater hardness. Important practical 
instances of the hardening effect of permanent set occur when 
plates or bars are rolled cold, hammered cold, or bent cold, or 
when wire is drawn. When a hole is punched in a plate the 
material contiguous to the hole is severely distorted by shear, and 
is so much hardened in consequence that when a strip containing 
the punched hole is broken by tensile stress the hardened portion, 
being unable to extend so much as the rest, receives an undue 
proportion of the stress, and the strip breaks with a smaller load 
than it would have borne had the stress been uniformly distributed. 
This bad effect of punching is especially noticeable in thick plates 
of mild steel. It disappears when a narrow ring of material 
surrounding the hole is removed by means of a rimer, so that the 



ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 43 

material that is left is homogeneous. Another remarkable instance 
of the same kind of action is seen when a mild-steel plate which is 
to be tested by bending has a piece cut from its edge by a 
shearing machine. The result of the shear is that the metal close 
to the edge is hardened, and, when the plate is bent, this part, 
being unable to stretch like the rest, starts a crack or a tear 
which quickly spreads across the plate on account of the fact that 
in the metal at the end of the crack there is an enormously high 
local intensity of stress. By the simple expedient of planing off 
the hardened edge before bending the plate homogeneity is re- 
stored, and the plate will then bend without damage. The injuri- 
ous effect of punching holes in thick plates of iron or steel is now 
so fully recognised that it is usual to specify that such holes shall 
be drilled. 

45. Annealing. The hardening effect of strain is removed 
by the process of annealing, that is, by heating to redness and 
cooling slowly. The effects of overstraining are got rid of by this 
treatment, and the material reverts to its primitive state. 

In the ordinary process of manufacture of iron or steel bars and 
plates by rolling, the metal generally leaves the rolls at so high a 
temperature that it is virtually annealed, more or less perfectly, and 
the behaviour of a sample in the commercial ^tate consequently 
does not differ much from that which the same sample would show 
if specially annealed*. The case is different with plates and bars 
that are " cold-rolled " or with pieces that have been hammered 
while in the cold state ; they exhibit the greater strength and 
much reduced plasticity which result from permanent set. A 
similar difference is found between wire supplied in the " hard- 
drawn " state and wire which has had the hardening effect of the 
last passage through the draw-plate removed by subsequent an- 
nealing. 

When pieces of a structure have been shaped by straining 
them while cold it is not unusual to anneal them afterwards In 
order to do away with the hardening effect of the overstrain. 

* In several of Mr Kirkaldy's papers a comparison is given of tin- clastic limit, 
ultimate strength, and ultimate extension of samples which were annealed befon 
testing, and of samples which were tested in the oommeroial state; In general the 
annealed samples are distinctly, though not very materially, softer than the Others. 
(on the Relative Properties of Wrought-Iron Plates from Essen and Yorkshire! London, 
1H70 ; also Experiments on Fayersta Stcl, I, on. Ion. 1S7M.) 



44 ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 

46. Hardening and Tempering of Steel. In wrought 
iron, very miid steel, and most other metals the rate of cooling 
from a red beat (in the process of annealing) is a matter of in- 
difference : much the same degree of softness is reached whether 
the cooling be fast or slow. But a slight difference may be 
observed with the mildest steel, and with steel containing more 
than 02 per cent, of carbon a marked difference is produced by 
fast as compared with slow cooling. When heated to bright 
redness and cooled suddenly, steel containing any considerable 
proportion of carbon is found to acquire hardness of a different 
kind from that which is produced by overstraining. A high-carbon 
steel chilled by plunging it at a cherry-red heat into cold water 
becomes so hard and brittle as to justify the title "glass-hard," 
which is sometimes applied to it. It is hard enough to scratch 
glass, and so brittle that a blow may break it into fragments. 

Steel treated in this way loses its plastic character entirely. 
When tested under tension it breaks with practically nothing but 
elastic extension, without contraction of section, and shows only a 
moderate amount of tensile strength. 

Further, the glass-hard steel may be deprived of its brittleness, 
have its strength increased, and have the range of' elastic strain 
greatly extended by subsequent heating to a moderate temperature. 
This process is called the tempering of steel. Its effect depends on 
the degree to which the temperature of the steel is raised, after it 
has been hardened. The different grades of temper which are 
produced in this way are often distinguished by reference to the 
colour (blue, straw &c.) which appears on a clean surface during 
the heating, in conseouence of the formation of a film of oxide. 

O ' J. 

Heating the hardened metal to a temperature between 400° F. and 
450 c F. produces a straw-coloured surface and develops a grade of 
temper suitable for the points of cutting tools intended to take a 
keen and hard edge ; a temperature of about 550° produces a 
purple-blue surface and gives a temper suitable for springs, where 
the desideratum is that the elasticity should be very perfect 
throughout a wide range of loads. When higher temperatures 
are used in the process of " letting dowm " the condition which is 
reached approaches more nearly to that of the annealed metal. 

47. Contraction of section at rupture. The extension 
which occurs when a bar of uniform section is tested by pull is at 



ULTIMATE STRENGTH AND NON- ELASTIC STRAIN. 



45 



first general, and is distributed with some approach to uniformity 
over the length of the bar. Before the bar breaks, however, a 
large additional amount of local extension occurs at and near the 
place of rupture. The material flows in that neighbourhood much 
more than in other parts of the bar, and the section is much more 
contracted there than elsewhere. The percentage contraction of 
area at fracture is frequently stated as one of the results of a 
test, and is a useful index to the quality of materials. If a 
flaw is present sufficient to determine the section at which the 
rupture shall occur, the contraction of area will in general be dis- 
tinctly diminished as compared with the contraction in a specimen 
free from flaws, although little reduction may be noted in the 
total extension of the piece. Local extension and contraction of 
area are almost absent in cast-iron and hard steel ; on the other 
hand they are specially prominent in wrought-iron, mild steel, 




Fig. 17. 

and other metals that combine plasticity with high tensile strength. 
An example is shown in fig. 17, which is copied from a photograph 
of a broken test-piece of Whit worth soft fluid-compressed steel. 
The piece was of uniform diameter before the test. 

48. Non-elastic Extension. Experiments with long rods 
show that the general extension which occurs in parts of the bar 
not near the break is somewhat irregular*; it exhibits here and 
there incipient local stretching, which has stopped without Leading 
to rupture. This is of course due in the first instance to want of 
homogeneity. It may be supposed that when local stretching 
begins at any point in the earlier stages of the tesl it is checked 
by the hardening effect of the strain, until finally, under greater 
load, a stage is reached in which the extension at one place goes 
on so fast that the hardening effect cannot keep pace with tin- 
increase in intensity of stress which results from diminution o( 
area; the local extension is then unstable, and rupture ensues, 

* See Kirkaldy's Experiments on Fagerata Steel, London, L878, also Report of tin- 
steel Committee, Part 1. 



46 ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 

Even at this stage a pause in the loading, and an interval of relief 
from stress, may harden the locally stretched part enough to make 
rupture occur somewhere else when the loading is continued. 

The molecular disturbance which occurs at the yield-point does 
not in general set in simultaneously in all parts of a test-piece 
however uniform the cross-section may be. The author has 
noticed it begin at one end, while along the greater portion of the 
length nothing but elastic extension was taking place, and then 
slowly spread from the place where it began until it included the 
whole of the piece, the load remaining constant all the while. The 
breakdown of structure appears to communicate itself gradually 
from place to place along the bar, each portion becoming in turn 
unstable when it finds itself deprived of support by the break- 
down of neighbouring portions. 

49. Crystalline Structure of Metals. Reference should 
be made in this connection to the results of microscopic exami- 
nation. When a metal is polished and lightly etched it is seen 
under the microscope to consist, in general, of crystalline grains, 
which are crystals with irregular outlines, the form of the 
boundaries having been determined by the meeting of the grains 
in their growth. Within each grain there is a definite orientation 
of the elementary pieces of which the crystal is built up, and the 
orientation changes from grain to grain. When the metal is 
stretched by pull, or by cold rolling or cold hammering, the grains 
become elongated. But when the piece, after the treatment, is 
heated to redness and is again examined microscopically, the 
grains are found to have re-formed and to be on the whole as 
long in one direction as in another. Slow cooling tends to produce 
large grains and fast cooling produces comparatively small grains. 

Microscopic observations by Mr W. Rosenhain and the author* 
have demonstrated that the manner in which a metal yields when 
it takes any kind of permanent set is by slips occurring on cleavage 
or " gliding " surfaces within each of the crystalline grains. These 
slips show themselves on a polished surface by developing systems 
of parallel lines or narrow bands, each of which is a step caused by 
one portion of the grain slipping over the neighbouring portion. 
Two, three, and even four systems of slip lines may be traced 

* Proc. Roy. Soc, March 16 and May 18, 1899. 



ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 47 

when the metal is considerably strained. Plasticity results from 
these slips, although the elementary portions of the crystals retain 
their primitive form and the crystalline structure of the metal 
as a whole is preserved. In some metals, in addition to simple 
slips or motions of pure translation, there is a molecular rotation 
resulting from strain, which gives rise to the production of " twin" 
crystals. Apart from this, however, the occurrence of slips on 
three or more planes within each grain suffices to allow the grain 
to change its form to any extent as the process of straining 
proceeds. 

50. Percentage of Extension in Tensile Tests. It is usual 
to measure the extension on a length, generally of 8 inches, in the 
middle portion of the piece under test, and to express the extension 
as a percentage of the length. Usually the fracture occurs within 
the length thus marked off, and when it does so the whole extension 
which is measured is partly general and partly local. The local 
extension, which occurs near the place of fracture, will affect the 
whole amount of extension to a degree that depends on the 
transverse dimensions of the piece as well as on its quality in 
respect of plasticity. A fine wire of iron or steel 8 inches long 
will stretch little more in proportion to its length than a very long 
wire of the same material, for with small transverse dimensions 
the local part of the stretching will be unimportant. On the other 
hand, a steel bar with a diameter say of 1 inch will show something 
like twice as much extension, in proportion to its length, as will 
be shown by a long rod. 

The experiments of M. Barba* show that, in material of uniform 
quality, the percentage of extension is constant for test -pieces of 
similar form, that is to say, for pieces of various size in which the 
transverse dimensions are varied in the same proportion as the 
length. It is to be regretted that in ordinary testing it is not 
practicable to reduce the pieces to a standard form, with one pro- 
portion of transverse dimensions to length, since an arbitrary 
choice of length and cross-section gives results which are incapable 
of direct comparison with one another. 

51. Forms of Test-Pieces. The form chosen for test pieces 

in tension tests affects not only the extension bul also the ultimate 

* Mrm. de in. Soc. des iiKi. Civ,, L880; Bee also a papei bj Mr \v. Baekney, "On 
the Adoption of Standard Forms of Test*Pieoea," Mm. Proc. Fml. C. /■"., L884, 



48 



ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 



strength. In the first place, if there is a sudden or rapid change 
in the area of cross-section at any part of the length under tension 
(as at AB, fig. 18), the stress will not be uniformly distributed 
there. The intensity will be greatest at the edges A 
and B, and the piece will, in consequence, pass its 
elastic limit at a less value of the total load than 
would be required if the change from the larger to the 
smaller section were gradual. In a non-ductile material, 
rupture will for the same reason take place at AB, 
with a less total load than would otherwise be borne. 
On the other hand, with a sufficiently ductile material, 
although the section AB is the first to be permanently 
deformed, owing to lack of uniformity in the distri- Fig. 18. 
bution of the stress there, rupture will preferably take 
place at some section not near AB, because at and near- AB the 
contraction of sectional area which precedes rupture is partly pre- 
vented by the presence of the projecting portions C and D. Hence, 
too, with a ductile material samples such as are sketched in fig. 19, 
in which the part of smallest section between the shoulders or 



*7p 



Mill"' "i™ 



D C 



O 



Fig. 19. 

enlarged ends of the piece is short, will break with a greater load 
than could be borne by long uniform rods of the same section. In 
good wrought-iron and mild steel the flow of metal preceding 
rupture and causing local contraction of section extends over a 
length six or eight times the width of the piece ; and, if the length 
throughout which the section is uniform be materially less than 
this, the process of flow will be rendered more difficult and the 
breaking load of the sample will be raised*. Forms of test-piece 
are to be preferred in which the length along which the section is 

* The greater strength of nicked or grooved specimens seems to have been first 
remarked by Mr Kirkaldy (Experiments on Wrought Iron and Steel, p. 74, also 
Experiments on Fagersta Steel, p. 27). See also a paper by Mr E.Richards, on tests 
of mild steel, Journ. Iron and Steel Inst., 1882. 



ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 49 

uniform is not less than 8 or 10 times the greatest transverse 
dimension, and if the piece has enlarged ends the change of section 
should not be quite abrupt. 

These considerations as to the choice of forms of test-pieces 
have a wider application than to the mere interpretation of special 
tests. An important practical case is that of riveted joints, 
in which the metal left between the rivet-holes is subjected to 
tensile stress. It is found to bear, per square inch, a greater pull 
than would be borne by a strip of the same plate, if the strip were 
tested in the usual way with uniform section throughout a length 
great enough to allow complete freedom of local flow*. 

52. Comparative Strength of Long and Short Rods. 

The tensile strength of long rods is affected by the length in quite 
a different way. With a perfectly homogeneous material, no 
difference should be found in the strength of rods of equal 
sectional area and of different lengths, provided the length of 
both were great enough to prevent the action described in § 51 
from affecting the result. But, since no material is perfectly 
homogeneous, the longer rod will in general be the weaker, offer- 
ing as it does more chances of a weak place ; and the probable 
defect of strength in the long rod will depend on the degree of 
variability of the material. When the degree of variability has 
been established by numerous tests of short samples, the strength 
which a rod of any assigned length may be expected to possess 
can be calculated by an application of the theory of probabilities. 
A theory of the strength of long bars has been worked out on 
this basis by Prof. Chaplinf, and has been experimentally con- 
firmed by tests of long and short samples of wire. The theory 
does not apply when the length is so small that the action of § 51 
enters into the case, and the experimental data on which it is 
based must be taken from tests of samples long enough to exclude 
that action. 



* See Kennedy's "Reports on Riveted Joints," Proc. Inst. Meeh, Eng. 1881-5. 
In the case of mild-steel plates a drilled strip may have as much as i-_> pel oent. 
more tensile strength per square inch than an umlnlled strip. With punched 
holes, on the other hand, the remaining metal is much weakened, for the reaton 
referred to in § 44. 

t Fan Nostrand'e Engineering Magazine, Dec. lssi); Proc. Engineers* club of 
Philadelphia, March, 1882. 

E, s. m. I 



50 ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 

53. Fracture by Tension or by Compression. In tension 
tests, rupture may occur by direct separation over a surface which 
is nearly plane and normal to the line of stress. This is usual in 
hard steel and other comparatively non-ductile materials. Or it 
may occur by shearing along an oblique surface. In very ductile 
samples these two modes of rupture are frequently found in com- 
bination, and the fractured surface is made up of a ceDtral core 
broken by direct tension while round it is a ring over which 
separation has taken place by shearing. The shorn ring often 
fomis a continuous cone or crater round a flat core. 

In compression tests of a plastic material, such as mild 
steel, a process of flow may go on without limit ; the piece, 
which must of course be short enough to avoid buckling^ 
shortens and bulges out in the form of a cask. This is illus- 
trated by fig. 20 (from one of Fairbairn's 
experiments), which shows the compres- 
sion of a circular cylinder of steel (the 
original height and diameter of which are 
shown by the dotted lines) by a load equal 
to 100 tons per square inch of original 
sectional area. The surface over which 
the stress is distributed becomes enlarged Flg - 20 - 

and the total load must be increased in a corresponding degree to 
maintain the process of flow*. The bulging often produces longi- 
tudinal cracks, as in the figure, especially when the material is 
fibrous as well as plastic (as in the case of wrought -iron). A brittle 
material, such as cast-iron, brick, or stone, yields by shearing on in- 
clined surfaces as in figs. 21. 22. which are taken from Hodgkinson's 
experiments on cast-iron i . The simplest fracture of this kind is 
exemplified by fig. 21, where a single surface (approximately a plane) 
of shear divides the compressed block into two wedges. With 
cast-iron the slope of the plane is such that this simple mode of 
fracture can take place only if the height of the block is not It--- 
than about one and a half the width of the base. When the 
height is less the action is more complex. Shearing must then 
take place over more than one plane, as in fig. 22, so that cones or 

* For examples, see Fairbainrs experiments on steel, Rep. Brit. Ass., 1S67. 
t Report of the Royal Commissioners on the Application of Iron to Railway 
Structures, 1849 ; see also Brit. Assoc. Rep., 1837. 




ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 



51 



wedges are formed by which the surrounding portions of the block 
are split off. The stress required to crush the block is conse- 





Fig. 21. 



Fig. 22. 



quently greater than if the height were sufficient for shearing in a 
single plane. 

54. Inclination of Surfaces of Shear in Tension and 

Compression Tests. The inclination of the surfaces of shear, 
when fracture takes place by shearing under a simple stress of 
pull or push, is a matter of much interest, throwing some light on 
the question how the resistance which a material exerts to stress 
of one kind is affected by the presence of stress of another kind, — 
a question scarcely touched by direct experiment. At the shorn 
surface there is, in the case of tension tests, a normal pull as well 
as a shearing stress, and in the case of compression tests a normal 
push as well as shearing stress. If this normal component were 
absent the material (assuming it to be isotropic) would shear in 
the surface of greatest shearing stress, which, as we have seen in 
§ 9, is a surface inclined at 45° to the axis. In fact, however, it 
does not shear on this surface. Hodgkinson's experiments on the 
compression of cast-iron give surfaces of shear whose normal is 
inclined at about 55° to the axis of stress* and Kirkaldy's, on the 
tension of steel, show that when rupture takes place by shear the 
normal to the surface is inclined at about 25 to the a\i> x . These 
results show that normal pull diminishes resistance to Bhearing 
and normal push increases resistance to shearing. In th< ifi 



Op. rit. 



t Op. rit. 



4—2 



52 ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 

cast-iron under compression, the material prefers to shear on a 
section whose normal is inclined at 55°, on which the intensity of 
shearing stress is only 0'94 of its value on the surface of maximum 
shearing stress, because on that surface the normal push is only 
066 of its value on the surface of maximum shearinp; stress. 

55. Fatigue of Metals. A matter of great practical as well 
as scientific interest is the weakening which materials undergo by 
repeated changes in their state of stress. It appears that in some 
if not in all materials a limited amount of stress-variation may be 
repeated time after time without appreciable deterioration in the 
strength of the piece ; in the balance-spring of a watch, for instance, 
tension and compression succeed each other some 150 millions of 
times in a year, and the spring works for years without apparent 
injury. In such cases the stresses lie well within the elastic limits. 
On the other hand, the toughest bar breaks after a small number 
of bendings to and fro, when these pass the elastic limits, although 
the stress may have a value greatly short of the normal ultimate 
strength. 

A laborious research by Wohler*, extending over twelve years, 
has given much important information regarding the effects on 
iron and steel of very numerous repeated alternations of stress 
from positive to negative, or between a higher and a lower value 
without change of sign. By means of ingeniously contrived ma- 
chines he submitted test-pieces to direct pull, alternated with 
complete or partial relaxation from pull, to repeated bending in 
one direction and also in opposite directions, and to repeated 
twisting towards one side and towards opposite sides. The results 
show that a stress greatly less than the ultimate strength (as 
tested in the usual way b} 7 a single application of load continued 
to rupture) is sufficient to break a piece if it be often enough 
removed and restored, or even alternated with a less stress of the 
same kind. In that case, however, the variation of stress being 
less, the number of repetitions required to produce rupture is 
greater. In general, the number of repetitions required to pro- 
duce rupture is increased by reducing the range through which 
the stress is varied, or by lowering the upper limit of that range. 

* Die Festigkeits-Versuche mit Eisen und Stahl, Berlin, 1870, or ZeiUchr. fur 
Bauwesen, 1860-70; see also Engineering, vol. xi., 1871. For early experiments by 
Fairbairn on the same subject, see Phil. Trans., 1864. 



ULTIMATE STRENGTH AND NON- ELASTIC STRAIN. 



53 



If the greatest stress be chosen small enough, it may be reduced, 
removed, or even reversed many million times without destroying 
the piece. Wohler's results are best shown by quoting a few 
figures selected from his experiments. The stresses are stated in 
centners per square zoll* ; in the case of bars subjected to bending 
they refer to the top and bottom sides, which are the most stressed 
parts of the bar. 



I. Iron bar in direct tension : — 



Stress. Number of Applications 
Max. Min. causing Rupture. 

480 800 

440 106,901 

400 .340,853 

360 480,852 



Stress. 
Max. Min. 

320 



440 200 



Number of Application* 
causing Rupture. 

10,141,645 



2,373,424 



440 240 Not broken with 4 millions. 



II. Iron bar bent by transverse load 



Stress. 
Max. Min. 



550 
500 



450 



Number of Bendings 
causing Rupture. 

169,750 
420,000 
481,950 



Stress. 
Max. Min. 



400 
350 



Number of Bendings 
causing Rupture. 

1,320,000 
4,035,400 



300 Not broken with 48 millions. 



III. Steel bar bent by transverse load 



Str 


3SS. 


Number of Bendings 


Stress. 


Number of Bendings 


Max. 


Min. 


causing Rupture. 


Max. Min. 


causing Rupture. 


900 





72,450 


900 400 


225,300 


900 


200 


81,200 


900 500 


764,900-mean of two trials 


900 


300 


156,200 


900 600 


Not broken with 33i mills 



IV. Iron bar bent by supporting at one end, the other end 
being loaded ; alternations of stress from pull to push caused by 
rotating the bar : — 



Stress. 


Number of Rotations 


Stress. 


Number of Rotations 


>m + to - 


causing Rupture. 


From + to - 


causing Rupture. 


320 


56,430 


220 


3,682,588 


300 


99,000 


200 


4,1)17.!'!'^ 


280 


183,1 15 


L80 


19,186,791 


260 


479,490 


L60 


Not broken with 


240 


909,810 




1 i'.^l millions. 



* According to Bausohinger (lor. ait., p. i»), the oentner per square loll In 
which Wohler gives his results is equivalent to 6*887 kiloa per Bquare om., or 0*048 I 

ton per square inch. 



54 ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 

From these and other experiments Wohler concluded that 
the wrought-iron to which the tests refer could probably bear an 
indefinite number of stress changes between the limits stated 
(in round numbers) in the following table. The ultimate tensile 
strength was about 19 J tons per square inch : — 

Stress in Tons per Sq. Inch. 

From pull to push +7 to -7 

From pull to no stress 13 to 

From pull to less pull 19 to 101 

Hence it appears that the actual strength of this material varies 
in a ratio which may be roughly given as 3 : 2 : 1 in the three 
cases of (a) steady pull, (b) pull alternating with no stress, very 
many times repeated, and (c) pull alternating with push, very many 
times repeated. For steel Wohler obtained results of a generally 
similar kind. His experiments were repeated by Spangenberg, 
who extended the inquiry to brass, gun-metal, and phosphor- 
bronze*. On the basis of Wohler 's results formulas have been 
devised by Launhardt, Weyrauch, and others to express the 
probable actual strength of metals under assigned variations of 
stress ; these are, of course, of a merely empirical character, and 
the data are not extensive enough to give them much value f. 
The general conclusions to which Wohler's experiments lead have 
been confirmed by the later researches of Sir B. Baker and 
Prof. BauschingerJ. They show how important it is to take 
account of the variability of the load in selecting a factor of 
safety. 

56. Imperfection of Elasticity. Wohler's experiments, 
dealing, as all experiments must deal, with a finite number of 
stress-changes, leave it an open question whether there are any 
limits within which a state of stress might be indefinitely often 
varied without finally destroying the material. It is natural to 
suppose that a material possessing perfect elasticity would suffer 
no deterioration from stress-changes lying within limits up to 

* Ueber das Verhalten der Metalle bei wiederholten Anstrengungen, Berlin, 1875. 

f See Weyrauch, " On the Calculation of Dimensions as depending on the 
Ultimate Working Strength of Materials," Min. Proc. Inst. C.E., vol. lxiii. p. 275 ; 
also a correspondence in Engineering, vol. xxix. ; and Unwin's Machine Design, 
chap. ii. 

X For details of the experiments bearing on the subject, see Prof. Unwin's book 
on The Testing of Materials of Construction, chap. xii. 



ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 55 

which the elasticity is perfect. But these limits, if they exist at all, 
are probably very narrow. Indeed, in the case of iron, there is 
indirect evidence that all alteration of stress whatsoever affects the 
molecular structure in a way not consistent with the notion of 
perfect elasticity. When the state of stress in iron is varied, 
however slowly and however little, the magnetic and thermo- 
electric qualities of the metal are found to change in an essentially 
irreversible manner*. Every variation leaves its mark on the 
quality of the piece ; the actual quality at any time is a function 
of all the states of stress in which the piece has previously 
been placed. It can scarcely be doubted that sufficiently refined 
methods of experiment would detect a similar want of reversibility 
in the mechanical effects of stress, even when alterations of stress 
take place very slowly. When they take place fast there is a 
want of reversibility due to another cause, pointed out by Lord 
Kelvin, namely, that the application of stress produces change of 
temperature, and consequently causes exchanges of heat to occur 
between the piece and its surroundings. Such exchanges of heat 
are necessarily irreversible, and hence the rapid application and 
removal of a load must do work on the piece even although a very 
slow application and removal of the load does no work. In other 
words, a material may be perfectly elastic in respect to indefi- 
nitely slow loading, and yet show dissipation of energy when the 
load is applied and removed somewhat quickly. Apart from this, 
however, there is probably in all materials some static hysteresis 
in the process of loading and unloading, corresponding to imper- 
fection of elasticity under indefinitely slow applications of loadf. 
And in stress-changes which occur rapidly, experiments show 
in general more dissipation of energy for strains within the 
usually accepted elastic limits, than can be accounted for by 
reference to the variations of temperature caused by straining. 
This is shown by the rate at which the vibrations of elastic solids 
subside. In experiments made by Lord Kelvin on the subsidence 
of the torsional oscillations of bodies suspended from wins, the 
bodies oscillating so as to twist the wire alternately to one and 
the other side, it was found that the rate of subsidence increased, 
in other words, the internal molecular friction causing imperfect 
elasticity increased from day to day when the who was kept 

* See papers by the author, /'////. ZVaiM., 1885, L886. 

t See experiments by the author. Rep. Brit. Assoc. L889, p. BOS. 



56 ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 

oscillating, but when the wire was allowed to rest for a day its 
elasticity improved. Thus it appears that repeated changes of 
stress have a cumulative effect in reducing elasticity, while 
Wohler's experiments show that they have also a cumulative 
effect in reducing strength. It may be conjectured that repeated 
strains induce a change in molecular structure of which the fatigue 
in strength and the fatigue in elasticity are two manifestations. 
A period of rest in a " fatigued " piece tends to restore elasticity : 
probably it would also tend to restore strength, but on this point 
experiments are as yet wanting. 

Annealing in any case serves to cure fatigue, and restores the 
primitive quality of the piece in respect of both strength and 
elasticity. 

It is remarkable that a piece which has been fatigued by 
many variations of stress, as in Wohler's experiments, and has had 
its endurance nearly exhausted, does not in general show any 
marked defect either in strength or in plasticity on being tested 
to rupture in the ordinary way. 

57. Cumulative Effect of Blows and Shocks. An effect 
which is sometimes confused with the phenomenon investigated 
by Wohler, but which should be treated as distinct from that, is 
the failure which is sometimes brought about through the 
cumulative effect of shocks. When a blow or shock expends 
kinetic energy in straining a piece the strain, which may be more 
or less general or more or less local according to the circumstances 
of the case, is such that the work done in producing it is equal to 
the energy of the blow. It may often happen that this exceeds 
the amount of work capable of being taken up in an elastic strain, 
and the limit of elasticity may, therefore, be passed in the strain 
to which some portion of the piece is subjected. This causes 
some local hardening, and as a similar effect may be frequently 
repeated the capacity of the piece to endure shocks may gradually 
become exhausted. A familiar instance is afforded by the chain of 
a winch, which in the course of protracted use may be exposed to 
many shocks from the slipping of the weight it lifts or from other 
causes. Any such shock may be said to use up a portion of the 
plasticity of the material, and the cumulative effect is to produce 
a hardening which might in time cause an unexpected failure if 
the chain were not periodically restored by annealing it. 



ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 57 

Fracture under successive blows, as in the testing of rails by 
placing them as beams resting on two supports, and allowing a 
weight to fall on the middle from a given height, results from the 
accumulated effect which is brought about in consequence of the 
energy of each blow being greater than can be absorbed by merely 
elastic strain. 

58. Initial Internal Stress. When stress is set up by 
applying an external load the behaviour of the piece may depend 
to a material degree on the existence of initial internal stress. A 
state of stress may exist while there is no external load. Taking 
some one section across the piece, we may for example have tensile 
stress occurring over some parts of the section, balanced so far 
as the resultant is concerned by compressive stress over other 
parts of the section. A piece which is entirely free from such 
actions is sometimes spoken of (in a phrase due to Prof. Karl 
Pearson) as being in a state of ease. Internal stress, existing 
without the application of force from without the piece, must 
satisfy the condition that its resultant vanishes over any complete 
cross-section. It may exist in consequence of set caused by pre- 
viously applied forces (a case of which instances are given below), 
or in consequence of previous temperature changes, as in cast-iron, 
which is thrown into a state of internal stress by unequally rapid 
cooling of the mass when it is being cast. Thus in a spherical or 
cylindrical casting an outside shell solidifies first, and has become 
partially contracted through cooling by the time the inside has 
become solid. The inside then contracts, and its contraction 
is resisted by the shell, which is thereby compressed in a tan- 
gential direction, while the metal in the interior is pulled in the 
direction of the radius. Allusion has already been made to the 
fact, pointed out by J. Thomson, that the defect of elasticity under 
small loads which Hodgkinson discovered in cast-iron is probably 
due to initial internal stress. In plastic metal a nearly complete 
state of ease is brought about by annealing; even annealed pieces, 
however, sometimes show, in the first loading, small defects of 
elasticity which may be due at least in part to initial stress, as 
they disappear or become reduced when the load is reapplied. 

59. Influence of Temperature on Strength Little is 

exactly known with regard to the effect of temperature Oil the 



58 ULTIMATE STRENGTH AND NON-ELASTIC STRAIN. 

strength of materials. Some metals, notably iron or steel contain- 
ing much phosphorus, show a marked increase in brittleness at low 
temperatures, or "cold shortness." Experiments on the tensile 
strength of wrought-iron and steel show in general little variation 
within the usual atmospheric range of heat and cold. The tensile 
strength appears to be slightly reduced at low temperatures, 
but to be practically unaffected through heating up to 100° or 
even 200° Fah. When the temperature exceeds 500° Fah. the 
tensile strength falls off rapidly, and at 1000° Fah. it is only 
a third or a fourth the normal value*. Reference may be made, 
in this connection, to the effect which a "blue heat," or tempera- 
ture considerably short of red heat, is believed to have on the 
plasticity and strength of iron, and more especially of mild 
steel. It appears that steel plates and bars bent or otherwise 
worked at a blue heat not only run a much more serious risk of 
fracture in the process than when worked either cold or red-hot, 
but become deteriorated in such a way that brittleness may 
show itself later when the metal is coldf. 

Prolonged exposure of wrought-iron or steel even to so mode- 
rate a temperature as 100° or 150° Fah., is known to bring about 
a gradual change of molecular structure, which shows itself in 
a marked variation of the magnetic quality of the metal, the 
primitive state, however, being reverted to when the metal is 
reannealed. It is not yet known whether any corresponding 
mechanical changes are brought about in this way in annealed 
metal, although, as we have seen in § 42, a brief exposure to such 
a temperature has an immense effect on a piece that has been 
newly overstrained. 

* See Report of a Committee of the Franklin Institute, 1837 ; Fairbairn, Brit. 
Assoc. Rep., 1856; Styffe on Iron and Steel, trans, by C. P. Sandberg. Notices of 
these and other experiments will be found in Unwin's Testing of Materials, in 
Thurston's Materials of Engineering, ii. chap, x., in Jour. Franklin Inst. 1881, vol. 
cxii. p. 241, and in papers by J. J. Webster, Min. Proc. Inst. C.E., vol. lx., and 
A. Martens, Zeitschr. des Ver. Deutsch. Ing., 1883. 

+ Stromeyer, "The Injurious Effect of a Blue Heat on Steel and Iron," Min. 
Proc. Inst. C.E., vol. lxxxiv., 1886. 



CHAPTER IV. 

THE TESTING OF MATERIALS. 

60. Testing Machines. In most modern testing machines 
the load is applied by means of hydraulic pressure acting on a 
piston or plunger to which one end of the specimen is secured, 
and the stress is measured by having the other end of the specimen 
connected to a lever or system of levers provided with adjustable 
weights. The hydraulic piston takes up the stretch as the test 
proceeds, doing work upon the test-piece, and the lever is kept 
floating by adjusting the weights on it as the stress increases. 
In many small machines and in some large ones screw-gearing is 
used instead of hydraulic power to apply force to the specimen. 

Springs are used in a few small machines (such as wire-testers) 
instead of weights, as a means of measuring the load. Another 
plan, which has been successfully employed even in large machines, 
is to make one end of the specimen act on a diaphragm forming 
part of a hydrostatic pressure gauge. 

61. Single-Lever Machine with vertically placed Test- 
Piece. A favourite and convenient form of testing machine, now- 
used in many English laboratories and steel-works is illustrated 
in fig. 23. This machine, designed by Mr J. H. Wicksteed, 
uses a single lever with a single heavy travelling weight as 
the means of measuring the force. 

The illustration shows a 50-ton machine, but machines of 
similar design have been built to exert a force o\' 100 tons or mere. 
The illustration includes an auxiliary apparatus for drawing auto- 
graphic diagrams of load and strain, which will be referred to later. 
The testing machine proper consists of a strong upright bo which 

the cylinder of the hydraulic ram is attached near the toot, and a 



60 



THE TESTING OF MATERIALS, 




□Q 






THE TESTING OF MATERIALS. 61 

long lever or weigh-beam resting by a knife-edge on the top of 
the upright, and carrying a travelling weight. The traveller 
weighs 1 ton ; it carries a vernier by which its position is read 
against a divided scale on the beam. The specimen is vertically 
placed : its lower end is fastened in a cross-head which is pulled 
down by the hydraulic ram below. The upper end is secured in 
a shackle which hangs from an inverted knife-edge on the beam. 
The beam itself oscillates about another knife-edge, at a short 
distance from the first, resting on the top of the supporting upright 
or " cat-head." The travelling weight is moved by means of a 
screw concealed within the beam, which takes its motion through 
spur wheels from a parallel shaft provided with a Hooke's joint in 
the axis of oscillation of the beam. This shaft is turned either by 
a handle wheel or by taking motion from a power-driven counter- 
shaft above — an arrangement which is chiefly convenient for 
running the weight back quickly after each test has been made. 
A counterpoise which is seen projecting behind the pillar near the 
ground serves to force up the ram when the hydraulic pressure is 
relieved. The pressure in the hydraulic cylinder is applied in 
some instances by means of a screw-pump, but a more convenient 
plan is to use an accumulator, or better still a " hydraulic in- 
tensifler" by which the work is done by admitting water from an 
ordinary low pressure main under a large piston. This large 
piston forces home a small hydraulic plunger, thereby producing a 
greatly increased intensity of pressure in the fluid on which the 
plunger acts, and that in its turn is transmitted to the straining 
ram of the machine. Such an apparatus has the advantage of 
allowing the load to be applied smoothly without shock, and at 
a slow or comparatively quick rate at will. The rate of application 
is regulated by a throttle valve through which water from the low- 
pressure main has to pass on its way to the large cylinder of the 
intensifies 

The machine is designed for tests in compression and bending 
as well as tension. Four columns, which are removable when tension 
tests only are to be made, connect the upper shackle with a plat- 
form in the shape of a cross-beam which hangs below the cross-head 
which is pulled down by the hydraulic ram. The arrangement is 
that of two stirrups linked with one another, one of t hem inverted, 
SO that when the two pull against, each other a block o( material 
placed between them becomes compressed. For tests m bending 



62 



THE TESTING OF MATERIALS. 




Fig. 24. Five-ton Testing Machine. 



THE TESTING OF MATERIALS. 63 

one of the stirrups — namely the beam which hangs by columns 
from the upper shackle — is made some four or five feet long and 
carries supports at its ends for the ends of the piece that is to be 
bent, while the cross-head presses down on the middle of the 
piece. In both cases the force which is exerted is measured by 
means of the weigh-beam and travelling weight, just as in tension 
tests. The arrangement for compression and bending tests will be 
clear from fig. 24, which shows a small machine by Mr Wicksteed 
capable of exerting a force of 5 tons, in which the force is exerted 
by means of a screw driven by hand. This machine has a supple- 
mentary arrangement for torsion tests. A worm and worm-wheel 
at the top of the upright serve to twist a specimen, one end of 
which is secured in the axle of the worm-wheel, while the other 
end is secured in a socket which projects from the side of the 
weigh-beam. The axis of the specimen under torsion is in the 
same line with the knife-edge about which the beam is free to 
oscillate. Each of the " knife-edges " in the weigh-beam of these 
machines is the edge of a square-cut bar of steel. The knife- 
edges are long enough to prevent the load on them from exceeding 
5 tons to the linear inch. With edges formed and proportioned 
in this way the friction is insignificant. Care has to be taken in 
testing to make the load come on smoothly and as far as possible 
to keep the beam from being thrown into oscillation, otherwise its 
inertia causes the maximum stress on the specimen to exceed the 
amount shown by the travelling weight. 

62. Calibration of Vertical Machines. Machines in which 
the specimen hangs vertically have the advantage of allowing the 
accuracy of their calibration to be readily tested by suspending a 
heavy weight of known amount from the upper shackle. The two 
points to be tested are (1) the distance between the knife-edges, 
and (2) the value of the travelling weight. The value of the 
travelling weight can be tested by the following method with- 
out removing it from the beam. Move the traveller until the 
beam stands horizontally midway between the stops. Then hang 
to a point of the beam at a known distance / from the fulcrum, 
and near the end, a known weight w. To balance the beam we 
must now move back the traveller through a distance /, which is 
measured by means of the scale and vernier. Then '/' the weight 
of the traveller is given by the equation 

Tl x = wl. 



64 



THE TESTING OF MATERIALS. 



In Mr Wicksteed's machines the weight of the traveller is 
generally 1 ton. It is conveniently checked hy hanging a 56 lb. 
weight to the beam at a point distant by 40 of the principal scale 
divisions from the fulcrum. The traveller should then require to 
be moved back one scale division to reestablish equilibrium. 

The distance between the knife-edges (on) is next determined 
by hanging a known heavy weight (say 1 ton) from the shackle 
and observing how far in scale divisions the traveller has to be 
moved to make the beam stand horizontal. Calling the weight W 
that is hung on, and l 2 the distance the traveller has to be moved, 
we have Wx=Tl 2 , 

an equation which determines x in scale divisions. Or, alternatively, 
when the weight W has been hung on, we may restore the equili- 
brium not by moving the traveller but by applying a measured 
(small) weight at some distant point on the beam, in the manner 
described above for testing the weight of the traveller. 

63. Other Testing Machines using Weights and Levers. 

For ordinary testing, in which the specimens to be dealt with are 
of no great length, nothing could exceed the convenience, 
accuracy and simplicity of the vertical machine. A preference 
however is in some cases felt for horizontal machines on account 
of the greater readiness with which they can be arranged to deal 
with exceptionally long test-pieces. The Werder testing machine, 
which is much used in Continental laboratories, is a horizontal 
machine with a single lever, so arranged that both the application 
and measurement of the load are effected at one end of the 
specimen, the other end being merely held fixed in the frame of 




Fig. 25. Scheme of Werder Testing Machine. 

the machine. The lever is of the bell-crank type with a short 
vertical arm and a long horizontal one and a fulcrum at the knee. 
The lever is pushed out bodily in a horizontal direction by the 



THE TESTING OF MATERIALS. 



65 



hydraulic ram : its short arm pulls one end of the specimen, and 
weights to measure the pull are applied to the long arm. The 
arrangement is shown diagrammatical ly in fig. 25. T is the piece 
under test, and L is the bell-crank lever to the long arm of which 
weight is applied while the ram R keeps this arm horizontal as the 
specimen stretches. 

Fig. 26 illustrates the general arrangement of a recent horizontal 
machine designed by Mr Wicksteed. Here tension tests are made 
in the space marked Q, and compression tests in either of the 




V7777777777777777* 

Fig. 26. Scheme of Wicksteed's Horizontal Testing Machine. 

spaces P or R*. The thrust of the hydraulic ram A is communi- 
cated through the test-piece to the suspended frame B, and thence 
through the bell-crank lever C and link D to the weigh-beam E. 
A view of the complete machine, taken from a photograph, is 
given in fig. 27. The cross-beam at the left-hand end is for 
bending tests. 

In many other testing machines a system of two, three, or more 
levers is employed to reduce the force between the specimen and 
the measuring weight. Probably the earliest machine of this 
class was that of Major Wadef in which one end of the specimen 
was held in a fixed support, and the stretch was taken up In- 
screwing up the fulcrum plate of one of the levers. In most 
multiple-lever machines, however, the fulcrunis arc fixed, and the 
stress is applied to one end of the specimen by hydraulic power 
or by screw gearing, which of course takes up the stretch, as in 
the single-lever machines already described. Kirkaldy, who was 
one of the earliest as well as one of the most assiduous workers 

* An account of this machine will be found in Engineering ^ •' ul \ . I 
t Report of Experiments on Metals /<>r Cannon, Philadelphia, L8 

e. s. M. 



66 



THE TESTING OF MATERIALS. 




THE TESTING OF MATERIALS. 67 

in this field, applied in his 1,000,000 lb. machine a horizontal 
hydraulic press directly to one end of the horizontal test-piece. 
The other end of the piece was connected to the short vertical 
arm of a bell-crank lever ; the long arm of this lever was horizontal, 
and was connected to a second lever to which weights are applied. 
In some of Messrs Fairbanks's machines the multiple-lever system 
is carried so far that the point of application of the weight moves 
24.000 times as far as the point of attachment to the test-piece. 
The same makers have employed a plan of adjusting automatically 
the position of the measuring weight, by making the scale lever 
complete an electric circuit w T hen it rises or falls so that it starts 
an electric motor which runs the weight out or in. Generally 
the measuring weight is adjusted by hand. In some, chiefly small, 
machines, the weight adjusts itself by means of another device. 
It is fixed at one point of a lever which is arranged as a pendulum, 
so that, when the test-piece is pulled by force applied at the other 
end, the pendulum lever is deflected from its originally vertical 
position and the weight acts with increasing leverage. 

Multiple-lever machines have the advantage that the measur- 
ing weight is reduced to a conveniently small value, and that it 
can be easily varied to suit test-pieces of different strengths. On 
the other hand, their multiplicity of joints makes the leverage 
somewhat uncertain and increases friction. 

64. Other Testing Machines. Diaphragm Machines. 

Hydraulic testing machines have been employed in which one 
end of the specimen is held in a fixed support and the stress is 
inferred from the pressure of the fluid in the hydraulic rani by 
which the load is applied, this pressure being read on a gauge. 
Machines of this class are open to the obvious objection that the 
friction of the hydraulic plunger causes a large and very uncertain 
■difference between the force exerted by the fluid on tin* plunger 
•and the force exerted by the plunger on the specimen. It appear-. 
however, that in the ordinary conditions of packing the friction 
is very nearly proportional to the fluid pressure, and its effect ma\ 
therefore be allowed for with some exactness. The method is 
not to be recommended for work requiring precision, unless the 
plunger be kept in constant rotation en its own axis during the 
test, in which case tin- effects of friction are almost entirety 

-eliminated. 






THE TESTING OF MATERIALS. 



In another important class of testing machines which we may 
distinguish as diaphragm machines, the stress (applied as before 

ne end of the piece, by gearing or by hydraulic pressure) is 
measured by connecting the other end to a free diaphragm 
which a liquid acts whose pressure is determined by a gauge. 
Fig. 28 shows a simple machine of this class 'used in 1873 
testing wire by Sir W. Thomson and the late Prof. Fleemin^ 




Kg. 28. Hydraulic Machine for Tcsrii:- Wire. 

Jenkin). The wire is stretched by means of a screw at the top. 
and pulls up the lower side of a hydrostatic bellows : water from 
the bellows rises in the gauge-tube G. and its height measures 
the stress 

In a larger testing machine of this type by Thomasset. the 
specimen pulls horizontally on the short end of a bell-crank lever, 
the long end of which presses on a horizontal diaphragm,, consisting 
of a metallic plate and a flexible ring of india-rubber. The 

--ure on the diaphragm displaces mercury from a chamber, 
which the diaphragm covers, and causes a column of mercury' to 
rise in a eausfe-tube therebv indicating the amount of the stress. 
The same principle is made use of in a number of other testing 
machines. It has found its most important application in the 
remarkable testing machine of Watertown arsenal, built in i s 7.' 
by the U.S. Government to the designs of Mr A. H. Emery. This 



THE TESTING OF MATERIALS. 



69 



is a horizontal machine, taking specimens of any length up to 
30 feet, and exerting a pull of 360 tons or a push of 480 tons by a 
hydraulic press at one end. The stress is taken at the other end 
by a group of four large vertical diaphragm presses, which com- 
municate by small tubes with four similar small diaphragm pre 
in the scale case. The pressure of these acts on a system of 
levers which terminates in the scale beam. The joints and bear- 
ings of all the levers are made frictionless by using flexible steel 
connecting plates instead of knife-edges. The total multiplication 
at the end of the scale beam is 420,000*. 

65. Testing Machines for special purposes. Small testing 
machines are made for such special purposes as determining 
the tensile strength of cement in briquettes, or the transverse 
strength of cast-iron bars, or for applying torsion. A usual test 
of cast-iron is to lay a rectangular bar with a section 2 inches 
deep and 1 inch wide on supports 3 feet apart, and load it in the 
middle until it breaks. The load required is usually between 
1 and 2 tons, and it is generally applied by means of a lever with 




Fig. 20. Arrangement of Levers in Torsion Testing Machine. 

a travelling weight. In torsion tests a worm and worm-wheel at 
one end of the specimen serve to apply twist, and the moment of 



i. tails of the Emery Machine ie€ "Report oft) 
L888, A;.] i i owin'fl I Materialt Kid Proe. Inst. M 

1888. 



70 THE TESTING OF MATERIALS. 

the couple may be measured at the other end either by a single 
loaded lever, or better by using a system of levers such as that 
sketched in fig. 29. The object in this arrangement is to secure 
that a pure couple will be applied. The lever AB, to which the 
specimen is secured at E. has equal arms. Half of the weight TT 
acts directly at B. The other half acts at B on one arm of the 
auxiliary lever CB. which is pivoted by a knife edge on a fixed 
support at its middle point F. This produces an upward thrust 
in the link CA equal to half the weight W, and hence the 
twisting moment on the specimen is constituted by a pair of 
equal and opposite forces forming a pure couple and escaping the 
shearing force which a weight simply applied at the end of a 
single lever would produce. 

66. Attachment of the specimens in tensile tests. The 

problem of holding a test-piece fairly so as to ensure that the pull 
will be symmetrically distributed about the axis, and that fracture 
will not occur at or near the grip through local inequality in the 
stress,, presents some difficulty,, especially when the material is of a 
rigid (non-plastic) kind. The shackles in which the piece is held 
are hinged, to let them adjust themselves to the line of pull. The 
test-piece is often made with enlarged ends on which screws are 
cut, and the end is screwed into a nut the seat of which is shaped 
to form part of a sphere, thus providing a ball-and-socket joint at 
each end of the bar. Or the enlarged end terminates in a shoulder 
inside of which two half rings are slipped on to form a collar, and 
the half rings have spherical curvature where they bear on the 
shackle. With plastic materials such as wrought-iron and mild 
steel there is no difficulty in getting a fair test of ultimate strength, 
even with the simplest appliances for holding the specimen, for the 
plastic vielding which precedes rupture wipes out any inequality 
there may be in the distribution of the stress to begin with. The 
trouble of screwing the ends or of forming shoulders on the test- 
piece may therefore be dispensed with, and a simpler attachment 
by wedge grips may be used. In this, which is the commonest of 
all methods of holding bars or strips of plate in commercial 
testing, each end of the bar stands between two wedges of hard 
steel the faces of which, where they press on the bar, are rough 
while the backs are smooth and are greased to make them slip 
down easily in a tapered recess in the shackle. When pull comes 



THE TESTING OF MATERIALS. 



71 



on the bar, the wedges are 
drawn down with it and 
press themselves against 
the specimen with so 
much force that the rough 
faces of the wedges bite 
into the plastic surface of 
the bar and hold it se- 
curely. In testing flat 
plate-strips the ends of the 
test-piece are usually cut 
a little wider than the 
main body of the piece, 
thereby giving an enlarged 
surface for the wedge to 
act on. With round or 
square bars no enlarge- 
ment of the ends need be 
used, but the wedges in- 
stead of being plane have 
a groove with roughened 
sides, so that each end of 
the bar is gripped at four 
places round its circum- 
ference. Fig. 30 shows 
in sectional elevation and 
plan the shackles of a 
Wicksteed 100-ton ma- 
chine with flat wedges 
holding a strip of plate 
as test-piece. The tapered 
hole in which the wedges 
sit is not cut out of a 
single piece of metal, but 
out of two half rings which 
are separately free to turn 
round the shackle, thus 
admitting of adaptation to 
Cases where the opposite 
sides of the strip are not 
quite parallel. 




Fig. BO. Bhaoklea in Wioksteed'a VertioaJ 
Testing Maohine. 



72 THE TESTING OF MATERIALS. 

67. Apparatus for drawing autographic diagrams of 
extension and load. In laboratory testing the relation of 

extension (beyond the elastic limit) to load throughout the test 
may readily be observed by simply applying a pair of beam 
compasses to two marked points on the specimen (usually 
8 inches apart) from time to time as the test proceeds, and 
transferring 1 the distance to a scale. When testing is to be done 
rapidly, and a knowledge of this relation is still wanted, some form 
of autographic recording apparatus is convenient. 

In most of the arrangements which have been designed for 

this purpose, the diagram is drawn by the relative movement of a 
pencil and a sheet of paper on a drum, one component of the 
motion being proportional to the extension and the other to the 
travel of the weight bv which the load is measured. In a single 
lever testing machine, for example such as that shown in fig. 23, 
a convenient form of recorder is made by supporting the paper 
drum horizontally on the main standard, setting the pencil 
carriage on a screwed spindle, which revolves along with the 
vertical shaft which gives motion to the travelling weight. 
This makes the pencil advance, parallel to the axis of the 
drum, through distances proportional to the load. The extension 
is taken by having two clips firmly secured to the test-piece 
at points 8 inches apart, with a fine wire or inextensible cord 
attached to one passing over a pulley on the other, thence over 
a second pulley on the first, and thence to the paper drum. 
This causes the dram to be turned round through distances 
proportional to the extension. In another arrangement, designed 
by Prof. Tin win*, the wire from the specimen causes a pencil to 
travel longitudinally, parallel to the axis of the drum, and the 
drum revolves through distances proportional to the displacement 
of the travelling weight. 

In Mr Wicksteed's hydraulic recorder, which appears on a 
separate frame behind the main standard in fig. 23, the drum 
is pulled round by a wire from the specimen, through distances 
proportional to the extension, and the pencil takes its motion, 
not from the travelling weight but from the piston of an 
auxiliary hydraulic cylinder in free communication with the 
straining cylinder of the machine. This piston compresses a 

* The Testing of Materials of Construction, Chap. vn. 



THE TESTING OF MATERIALS. 73 

spring in its advance, and therefore its displacement measures the 
force with which it is pressed out. Its friction is eliminated by 
keeping it in continuous rotation, and this makes it indicate 
correctly the pressure in the main straining cylinder. But the 
net load on the bar bears a somewhat uncertain relation to that 
pressure, in consequence of the friction of the ram. Mr Wicksteed 
contends that the friction of the ram is proportional to the pressure, 
and that a uniform scale is therefore found, the value of which is 
interpreted by occasional reference to the weigh-beam. 

68. Measurement of Young's Modulus by Extenso- 
meters. The small strains which occur in a tensile test of 
non-plastic material, and those which occur during the early 
stages of the test in material of any kind, require some form of 
delicate measuring appliance. The name extensometer is given 
to apparatus designed for this purpose. We have seen that the 
whole amount of elastic stretching in such a material as wrought- 
iron amounts to only about yoVo" oi> the l en gth under observation. 
In measuring the elastic modulus and in determining the true 
elastic limit we must be able to compare the fractional parts of 
this strain which are produced by successive increments of load. On 
an 8-inch length of iron or steel the elastic extension is, in round 
numbers, about T gVo mcn f° r eacn ton per square inch of load. 
Hence to obtain accurate measurements of the modulus there is 
much advantage in being able to read to, say, ^wu °f an incn - 

Measurements taken between marks on one side of the bar are 
so much affected by any bending of the bar through accidental 
inequality in the distribution of the stress that no credit is to be 
given them. It is essential either to measure the extensions on 
opposite sides of the bar and take a mean of the two, or to measure 
the displacement between two pieces which are attached to the 
bar in such a way as to share equally in the strain on both Bides. 

In the experiments of Bauschinger, which take high rank among 
observations of this class, independent measurements of the strains 
on two sides of the bar were taken by using mirror micrometers of 
the type illustrated in fig. 31. There are two dips a and t> clasp- 
ing the test-piece at the places between which th«' extension is bo 
be measured. The clip b carries two small rollers, a\, </,. which are 
free to rotate on centres fixed in the clip. These rollers press on 
two plane strips attached t<> the other clip. When the specimen 



74 



THE TESTING OF MATERIALS. 



stretches the rollers are consequently caused to turn through dis- 
tances proportional to the strain. The amounts by which the 



""^ — 



*z. 




U-- *-" 



Fig. 31. Scheme of Bauschinger's Extensometer. 

rollers turn are read by means of small mirrors, g lt g 2 , fastened to 
the rollers, which reflect the mark- 
ing of a fixed scale, f, into the 
reading telescopes e l3 e 2 . The ex- 
tension of the bar is deduced from 
the mean of the two readings. The 
adjustment of this apparatus is a 
matter of considerable nicety, and 
in point of convenience a self-con- 
tained form of extensometer is much 
to be preferred. 

Professor Un win's extensometer 
(fig. 32) uses two clips, c lt c. 2 , the 
upper one of which is free to revolve 
about the pair of points which attach 
it to the bar, while the lower one is 
deprived of this freedom by a set 
screw s abutting on the side of the 
bar. Each of the clips has a level 
tube I fastened to it, and the lower 
one carries a rod r furnished with a 
micrometer screw m, the point of 
which presses against the under side 
of the upper clip. The screw s is 
adjusted to make the lower level 
tube horizontal : then the upper tube 

is Set level also by adjusting m, and Fig. 32. Unwin's Extensometer. 




THE TESTING OF MATERIALS. 



75 



as the specimen stretches the amount is noted by which m has 
to be turned to keep the tube level. The instrument reads to 



tofo o inch - 



The author has devised an extensometer of the self-contained 
class which has proved convenient and accurate in use. It can be 
quickly applied to any test-piece and no part of it has to be 
touched while the test is being made. The principle involved 
is illustrated diagrammatically in fig. 33. There are two clips B and 




©c 




olQ 



0B 



Fig. 33. Scheme of the Author's Extensometer. 

G each attached to the test piece A by the points of two set-screws. 
The clip B has a projection B' ending in a rounded point P which 
engages with a conical hole in 0; when the bar extends this 
rounded point serves as a fulcrum for the clip C and hence a point 
Q, equally distant on the other side, moves relatively bo B through 
a distance equal to twice the extension. This distance is measured 
by means of a microscope attached to B either on a projection 
which allows it to point directly towards a mark at Q t or ta- 
in the sketch) the microscope forms a prolongation of B ami 
the motion of Q is brought into the held of view by means 



76 



THE TESTING OF MATERIALS. 



of a hanging rod R. The rod R is free to slide on a guide 
in B, and carries a mark on which the microscope is sighted. 
The displacement is read by means of a micrometer scale in 
the eye-piece of the microscope. The pieces B and B' are jointed 
to one another in such a way that the bar may twist a little, as it 
is sometimes liable to do during a test, without affecting the 
engagement of P with C. This also obviates any need of absolute 
parallelism in the axes of attachment of the two clips. But the 
joint between B and B' forms a rigid connection so far as angular 
movement in the plane of the paper is concerned. This feature is 
essential to the action of the instrument : it is only then that P 
serves as a fixed fulcrum in the tilting of A by extension on the 
part of the specimen. 

Fig. 34 is a view of one form of the complete instrument, taken 




Fig. 34. The Author's Extensometer. 



THE TESTING OF MATERIALS. 77 

from a photograph*. The clips B and C are set at 8 inches apart. 
Here the instrument is inverted, as compared with the scheme of 
fig. 33. The joint between B and B' consists, in this instance, of two 
upright pins fixed in B, one of which presses up into a hole and the 
other into a slot in B', the line of this hole and slot being perpen- 
dicular to the axis of the set-screws by which the clip is attached 
to the rod under test. Hence, so far as movement about the axis 
of the set-screws is concerned, B and B' act as a rigid whole. This 
movement is prevented by the gearing of P in the hole in the 
lower clip C. The piece B' is here a frame consisting of two 
parallel steel rods united by a cross-bar at top and bottom, and 
carrying, besides the screw P, the microscope, which is hinged 
to B' about the point E vertically above Q, and is provided with a 
focussing screw at F. The counterpoise D, which is also attached 
to the piece B', serves to balance the weight of the microscope 
and make the pressure vertical between P and the hole into which 
it gears. There is a supplementary counterpoise D' for adjusting 
the balance about the axis of the joint between B and B'. These 
counterpoises are adjusted so that when the heavy end (Q) of C is 
raised, making P cease to be in gear with C, P has no tendency to 
move in any direction. The excess of weight on the right-hand 
side of G may be made sufficient to produce the requisite pressure 
at the point P but it is convenient to supplement this pressure by 
means of a light spring pulling the two together. The frame BB' 
with the microscope may be lifted off, leaving only the two clips 
attached to the rod. 

The object sighted is one side of a wire stretched horizontally 
across a hole in a plate at Q, and illuminated by a small mirror 
behind. The distances OP and OQ are in this instance equal, with 
the effect that the movement of Q is double the extension of the 
rod. The length of the microscope is adjusted, with reference to 
the scale in the eye-piece, so that the numbers read on the scale 
correspond to -gTrJiju °f an mcn °f extension. This adjustment is 
tested by turning the screw P, which has a pitch of .,',, inch, 
through one revolution, and observing that the displacement of 
Q is 500 units of the eye-piece scale. In the instrument illus- 
trated in fig. 34 the whole scale comprises 1,400 units, and 
calibration tests show that throughout the middle L,200 of them 

* Proc. Roy, Society, Vol. 68, M«iy. L896". 



78 



THE TESTING OF MATERIALS. 



the proportionality of the scale readings with the real movements 
of Q is practically perfect. 

The scale engraved in the eye-piece of the microscope has 140 
divisions each corresponding to 50V0 mcn °f extension, and by 
estimation to tenths of a division readings are taken to 5-0^00 inch. 

The screw P further serves to bring the sighted mark to a 
convenient point on the micrometer scale, and also to bring the 
mark back if the strain is so large as to carry it out of the field of 
view : thus a single turn of the screw adds 500 scale divisions to 




Fig. 35. The Author's Extensometer (Newer Form). 

the range shown on the micrometer scale. In dealing with elastic 
strains there is no need for this, as the range of the scale is itself 



THE TESTING OF MATERIALS. 79 

sufficient to include them, but it is useful when observations are 
being made on the behaviour of metals as the elastic limit is 
passed. 

To facilitate the application of the extensometer to any rod a 
clamp (not shown in the figure) is added by which the clips B and 
C are held at the right distance apart with the axes of their set- 
screws parallel, while they are being secured to the test-piece. 
Such a clamp is especially convenient when the strain has been 
carried beyond the elastic limit and it is desired immediately to 
reset the clips to the standard distance apart after the length 
between them has materially changed by the extension of the 
specimen. 

The form shown in fig. 34 is applicable to vertical specimens 
only. A newer and in some respects more simple form, suitable 
for horizontal or inclined bars as well as for vertical ones, is shown 
in fig. 35. In this arrangement the microscope forms a prolonga- 
tion of the clip B, and the displacement of the point Q in the clip 
C is brought into the field of view by a rod R, as in the scheme 
sketched in fig. 33. A ball-and-socket joint is used between R and 
C, and the two are held together by a light spring. The calibrating 
screw is now fixed in C and a hole at the end of it forms the socket 
for B'. 

Fig. 36 shows what is substantially the same form of extenso- 
meter adapted to measure the elastic compression of short blocks. 




Fig. 86. The Author's Extensometer applied to measure the Elastic 
Compression <>f short Blooks. 

Here the length t<> be dealt, with i».'i ween the <-lips is only l \ inches, 
and the strain of the specimen is mechanically multiplied LO times 



80 



THE TESTIXG OF MATERIALS. 



instead of only twice, as in the former case. This is done by 
extending the clips to the right so that the distance of Q from 
the axis is 9 times that of P. The prolongations to the left are 
added to counterpoise the weight, so that the force with which 
the point P presses against its socket may be vertical. The 
motion of Q is transferred to the field of the microscope by means 
of a vertical hanging piece which is jointed to the lever PQ 
at Q, and carries the mark on which sights are taken. In this 
instrument the calibrating screw is dispensed with, and the object 
sighted by the microscope is a small piece of glass on which two 
fine horizontal lines are engraved at a distance of J- inch apart. 
The length of the microscope is adjusted to make these lines 
include 500 units of the eye-piece scale. Each unit consequently 
corresponds to a displacement of the glass plate' through 25 ^ QQ inch, 
or in other words to an extension in the test-piece of 9-50W0 i ncn - 

Extensometers such as have been described are an important 
adjunct to the testing machine, and are most commonly used 
along with it. It is however worth while to observe that for 
many experiments on elasticity the testing machine is not essential. 




Fig. 37. Small Testing Machine for Elastic Extension of Eods, 
with Extensometer attached. 



THE TESTING OF MATERIALS. 81 

So long as elastic strains are dealt with there is no need of 
hydraulic or other gearing to take up the stretch, and a simple 
lever may suffice to apply the load. A laboratory apparatus for 
measuring Young's modulus in rods of various metals is shown in 
fig. 37. By hanging weights from the end of the lever loads up 
to 1 ton are applied to the rod, which may conveniently have a 
diameter of J or | inch, and an extensometer attached to the rod 
measures the strain. 

69. Measurement of Young's Modulus in Wire. When 

long pieces of wire are available the elastic extension admits of 
direct measurement by means of a scale and vernier. A good 
plan is to hang up two long wires side by side, fastening both to 
the same support and attaching the scale to one and the vernier 
to the other. One is kept taut by a load which is not varied 
during the test. The other is first loaded with a weight sufficient 
to straighten it, and additional weights are then applied to produce 
the elastic extension, which is measured by noting the movement 
of the vernier over the scale. With iron or steel wires, say 20 feet 
long, the extension will be nearly -^ inch for each ton per square 
inch of load, and as the load may generally be raised to 10 tons 
per square inch, and often to much more without passing the 
elastic limit, the movement of the vernier is sufficient to give 
fairly accurate determinations of the modulus. The advantage 
of using a second wire to carry the scale is that any yielding 
of the support, or any change of temperature such as might 
occur during the test, affects both wires equally. 

When comparatively short pieces of wire are used some means 
of magnifying the relative displacement is necessary. A con- 
venient plan is to clamp two little blocks to the two wires to 
serve as platforms on which is placed a small tripod carrying a 
mirror, two feet of the tripod being supported in a hole and a slot 
respectively on one of the blocks while the other leg rests on a 
plane horizontal surface on the other block. When one wire 
stretches the mirror tilts, and the amount of its tilting is measured 
by means of a fixed reading telescope and scale. Fig. 38 shows 
an apparatus for carrying out such measurements. The wires 
hang inside a tubular stem from a clamp at the top, and a cross- 
bar attached to the bottom end of one of thom carries a constant 

quantity of load \yhile variable load is applied to the other. The 
k. s. m. 6 



82 



THE TESTING OF MATERIALS. 



reading telescope with its attached scale is supported by part 
of the framework so that the whole apparatus is self-contained. 
A fixed shelf may be used in place of the second wire. 




Fig. 38. Apparatus for measuring elastic extension of wires. 



In calculating the extension from the scale readings it must be 
noted that the angle through which the reflected ray turns is 
twice the angle through which the mirror tilts. Let a be the 
effective width of the tripod carrying the mirror, namely, the dis- 
tance from its back foot to the line joining its two front feet, let s 
be the number of scale divisions by which the reading in the 
telescope changes when a load is applied, and let b be the distance 

between the mirror and the scale expressed in scale divisions. 

g 
Then the small angle turned through by the ray is j . The angle 



THE TESTING OF MATERIALS. 



83 



through which the mirror tilts is — , where 81 is the extension of the 

ci 

wire. Hence 8l = —r and is found in the same units as are used in 
2b 

measuring a. 

70 Measurement of Young's Modulus by Bending 

Observations of the deflection of a loaded bar supported as a beam 
on fixed supports, or clamped at one end and free at the other, 
furnish a convenient method of determining Young's modulus for 
the material of the bar. When the piece is long and sufficiently 
flexible to bend considerably, the deflection is readily measured by 
having a fixed scale behind the beam, with a fixed piece of mirror 
glass alongside of the scale, so that readings may be directly taken 
by bringing the eye to the level of the beam until the top edge 
of the beam covers its reflexion in the mirror behind, and then 
sighting the position of the edge upon the scale. In dealing with 
less flexible bars an apparatus like that shown in fig. 39 is useful. 




Fig. 39. Apparatus for measuring elasticity by deflection of beams. 



There the supporting knife-edges arc clamped on a stiff frame 
Like a Lathe-bed, and can be Bet to any desired distance apart. 

6 2 



84 THE TESTING OF MATERIALS. 

The deflection is measured by sighting a finely divided glass scale, 
clamped to the beam, through a reading microscope of low power. 
A useful addition to the apparatus consists of a little mirror which 
can be set astride the beam at any place for the purpose of 
observing the angle of slope there, the tilting of this mirror when 
the beam is loaded being observed from a distance by means of a 
reading telescope and scale. This mirror appears in the figure above 
one of the two supports. In place of loading in the centre, two loads, 
equal in amount, may be applied at the two extremities of the 
beam, which are arranged for that purpose to project by equal 
distances beyond the two supports. The advantage of this method 
of loading is that the middle portion of the beam between the 
supports is then subject to uniform bending and to no other kind 
of strain — a point which will be explained in the chapter dealing 
with beams. 

Let a be half the distance between the supports, and b the 
distance by which the beam projects beyond each support. It will 
be shown later (Chapter VII.) that if a load W be applied at 
the centre the deflection caused then by that load is 

Wa 3 



6A7 



where / is the moment of inertia of the section about a horizontal 
central axis. 



Hence in that case 



E- — 



Again, if a load W be applied at each extremity, the upward 
deflection at the centre is given by the equation 

Wa*b 

and in that case 

Wa'b 



E = 



2u,I 



Fig. 40 shows a similar arrangement for observing the deflec- 
tion of a cantilever or beam held fixed at one end and free at the 
other. Taking L to denote the whole length from the clamp to 



THE TESTING OF MATERIALS. 



85 



the free end, and assuming the load W to be applied at the free 
end as in the illustration, the deflection there is 

WD 






SET' 




Fig. 40. Apparatus for observing deflection of cantilever. 

The practical difficulty of ensuring that the clamp shall hold 
the fixed end so securely as to keep it strictly horizontal makes 
this experiment a less trustworthy means of finding E than the 
other. 

In both cases the apparatus is arranged so that the deflection 
may be observed at various points along the length of the bar for 
the purpose of examining experimentally the curve which a beam 
of uniform section assumes under a given load or system of loads. 
The slope may also be determined from point to point along the 
length by means of the jockey mirror. 

71. Measurement of the Modulus of Rigidity. Static 
Method. This modulus is usually measured by experiments on 
the torsion of a round rod or wire. It will be shown later that 
when such a rod is twisted every part of it is in a state o[' shear. 
and that within the clastic limit the angle of twist B (expressed in 
circular measure) on any length / is connected with the diameter 



86 



THE TESTING OF MATERIALS. 



d, the modulus of rigidity C, and the twisting moment M by the 

equation 

n S21M 

u = 



or 



C = 



S21M 



7rd 4 0' 

In applying this to measure C in rods of moderate diameter a 
convenient plan is to find 6 by 
using two long pointers, clamped 
to the rod near its ends with their 
distant ends moving over fixed 
scales. The difference of the 
two scale readings measures the 
twist on the length I between the 
pointers. When the diameter of 
the rod is so great as to make the 
angle of twist too small to be 
measured in this way, a pair of 
mirrors clamped on the rod and 
facing sideways should be used 
along with a reading telescope and 
scale for each. An optical pointer 
has the advantage over a mechani- 
cal pointer of doubling the angle, 
and further it can readily be made 
of much greater length. 

Fig. 41 shows a self-contained 
apparatus for experiments on the 
torsion of wires. The wire hangs 
in the axis of a tubular stem and 
carries a cylindrical weight round 
which two cords pass which are led 
away over pulleys and carry hangers 
on which equal weights are placed. 
The wire is consequently twisted 
by a pure couple, and the angle of 
twist is read by observing the dis- 
placement on a fixed circular scale 
of a pointer attached to the cylin- 
drical weight. 




Fig. 41. Apparatus for measuring 
modulus of rigidity by the tor- 
sion of wires. 



THE TESTING OF MATERIALS. 87 

72. Measurement of the Modulus of Rigidity. Kinetic 
Method by Torsional Oscillations. Let a circular rod or wire 
be held fixed at one end and have attached rigidly to the other 
end a mass which is set into oscillation by applying a twist and 
letting go. Then for elastic twists the moment acting at any 
instant on the mass to restore it to its normal position will be 
proportional to the angle of twist at that instant, and the oscilla- 
tions will consequently be of the simple harmonic type, and will be 
executed in the same period whether they are large or small, 
provided they lie within the elastic limit. Thus if t is the period 
of time taken to make each complete oscillation, and /u is the 
twisting moment per unit of angle (in other words, the constant 
ratio of the twisting moment M to the angle 6), we have 

t = 27T\/ , 

when I is the moment of inertia of the oscillating mass about the 
axis of the rod. The factor g converts the moment \x into kinetic 
units. 

But by the principle stated in § 71 (and to be proved later) 



Hence t 2 = 

fig gd*C 



and C = 



.1/ ird'G 

±ir 2 I _ V2SirlI 
128irll 



gdW 



which allows G to be found by observing the period t, when the 
diameter and length of the rod and the moment of inertia of the 
oscillating mass are known. 

The apparatus shown in fig. 41 allows this moans oi' measuring 
C to be carried out on the same wire to which the static test is 
applied. The cords have simply to be disconnected and the heavy 
cylindrical mass to be set oscillating. Its moment of inertia is 

9 , r being its radius and m its mass. 

Another convenient \'<>vu\ of oscillator consists of a hollo* 



88 



THE TESTING OF MATERIALS. 



cylinder or ring with a rectangular bar across the top to allow the 
wire to be attached, fig. 42. Its mo- 
ment of inertia is 

??z 1 (?V" — ?'2 2 ) m. 2 {a- 4- b 2 ) 

where m^ is the mass of the ring i\ and 
r.> its external and internal radii; nu is 
the mass of the bar. a its half length 
and b its half breadth measured hori- 
zontally. 

The method of oscillations may be 
used with rods of considerable dia- 
meter by attaching a cross bar to which 
heavy masses may be applied. The 
moment of inertia is most readily found 
by noting the period when the amount 
of the applied masses is changed. Thus after observing the 
period t lf let an additional mass. m, be applied at each end of 
the cross bar, and let t. 2 be the value then found for the increased 
period of oscillation. Calling the original moment of inertia /, 
the moment of inertia in the second state is I+2ma 2 , where a is 
the distance from the axis to the points at which the weights are 
applied, and 

from which 




Fie. 42. 



/ = 



I + 2ma 2 



73. Maxwell's Needle used as Torsional Oscillator. 

This is a particularly convenient oscillator for use in measuring 

It consists (fig. 43) of a tube 



the modulus of rigidity of wire 



Fie. 43. 



Fie. 43 a. 



Maxwell's Xeedle. 



into which four equal short pieces of tube can be slipped, each of 
the short pieces being one-fourth of the length of the long tube. 



THE TESTING OF MATERIALS. 



89 



Two of the short pieces are empty and two are filled with lead. 
By placing the tubes as shown in figs. 43 and 43 a the moment of 
inertia of the system can have two values given to it, I x and 
I 2 , of which I x (corresponding to fig. 43) is considerably the 
greater. 

To express the change in moment of inertia, or 1^ — I 2 , let a be 
the half length of the long tube, m 1 the mass of each of the two 
short tubes that are filled with lead, and m 2 the mass of each of 
the empty short tubes. 

Then the system is changed by shifting two masses each equal 
to m x — m 2 so that the distance of the centre of gravity of each 
from the axis changes from fa to \ a. 



Hence 



I 1 — I 2 = 2 (??*! — m 2 ) (jqCi 2 — jjrd 2 ) = (m x — m 2 ) a 2 . 




Pig. 41. Apparatus for experiments on torsional oscillation, 
usinj^ Maxwell's Needle. 



90 THE TESTING OF MATERIALS. 

Let ti and t 2 be the observed periods of oscillation in the two 
cases respectively. 

Then 

tf — t 2 - 7 X — I 2 (m 1 — m 2 ) cC 2 ' 
I 1 (m 1 — m 2 ) a 2 



or 



Cj Zi" ~~ L 2 



But C = ^A 

gd* t ± 2 ' 

and hence we obtain without calculating / the following equation 
for C, 

p _ 1287r£ (m 1 — m 2 ) a 2 
~ gd± ' t; 2 - U 2 ' 

A self-contained apparatus for experiments on the torsion of 
wires by means of Maxwell's needle is shown in fig. 44. 

74. Results of Tests. Data for Cast-iron. Cast-iron, 
the product of the blast furnace, has properties which vary widely 
in different specimens, depending as they do in great measure on 
the quantity of carbon which the iron contains, as well as on the 
manner in w 7 hich the carbon is united to the iron. The amount 
of carbon may range from 2 up to nearly 5 per cent. In white 
cast-iron it exists mainly in a state of combination with the iron : 
in grey cast-iron it consists mainly of graphite mixed with the 
iron. Silicon is also present in amounts that vary from less than 
1 up to 3 per cent, or even more, along with small quantities of 
sulphur, phosphorus and manganese. Comparatively great softness 
is obtained when the amount of carbon present in the combined 
state is small, although the whole amount of carbon may be con- 
siderable, but a much stronger iron is obtained by having 1 per 
cent, or more of carbon in the combined state. 

The tensile strength of cast-iron may be as low as 4 tons 
per square inch, and may be as high as 20 tons. These are 
exceptional figures, and values ranging between 8 and 12 tons per 
square inch are more usual in good foundry iron. The compressive 
strength may be as low as 20 tons per square inch, and as high as 
nearly 100 tons in exceptional cases : its ordinary values range 



THE TESTING OF MATERIALS. 91 

from 40 to 60 tons per square inch. The shearing strength 
appears from such experiments as have been published to be 
somewhat lower than the tensile strength. The great compressive 
strength of cast-iron leads to its being largely used in the con- 
struction of columns, but the facility with which it can be melted 
and cast into any desired shape is the property to which its 
application in engineering is mainly due. Very grey cast-iron is 
the kind which becomes most perfectly fluid when melted, and 
consequently takes most exactly the form of the mould, but a 
less grey mixture is to be preferred when strength is a chief 
desideratum. 

It is only within very narrow limits that cast-iron can be 
said to show even approximate proportionality between stress 
and strain. Hodgkinson's experiments on long cast-iron bars 
showed that both in extension and in compression the strain was 
about '00017 for each ton per square inch of load, in the initial 
stages of the loading. This makes, in round numbers, 

E — 7 = 6000 tons per sq. inch. 

In experiments on short cast-iron bars Prof. Unwin* observed 
strains ranging in various specimens from '000133 to '000156 per 
ton per square inch. The corresponding limits of E are in round 
numbers 7,500 and 6,400 tons per square inch. Values of the 
modulus of rigidity C in cast-iron generally lie between 3,400 and 
3,000 tons per square inch. 

75. Wrought-iron. In wrought-iron, which is manufactured 
from cast-iron by the processes of puddling and rolling, only a 
small fraction of the carbon present in the cast-iron survives the 
action of the puddling furnace. The carbon remaining in the iron 
is reduced to less, often to much less, than one-quarter of one per 
cent. Traces of manganese, sulphur, silicon and phosphorus are 
also found. The metal as rolled has a markedly fibrous structure, 
arising chiefly from intermixture with particles of slag which the 
process of rolling draws out into long filaments. 

Good wrought-iron bars have a tensile strength of about '2~> 
tons per square inch, and will stretch as much as -<> per cent on 
an 8-inch length before breaking, the section becoming lvduoed 

* 77//' Testing of Materials of Construction, 1st Ed, p. '-'' 



92 THE TESTING OF MATERIALS. 

at the place where fracture occurs to 50 or 60 per cent, of its 
original size. In good wrought-iron plates the strength of strips 
cut along the direction of rolling ranges generally from 20 to 24 
tons per square inch, and that of strips cut across the direction 
of rolling from 18 to 22. In plates of poor quality the tensile 
strength may be as low as 16 tons per square inch. Bars having 
a tensile strength of say 23 or 25 tons have usually a well-marked 
yield-point in the neighbourhood of 15 or 17 tons. The crushing 
strength of wrought-iron is rather indefinite : it is often taken as 
| of the tensile strength. The shearing strength ranges from about 
16 to 20 tons per square inch, but is according to Bauschinger's 
experiments notably less when a plate is sheared in a plane parallel 
to its faces. In that direction the shearing strength may be only 
8 or 10 tons. The value of E for wrought-iron lies in most cases 
between 12,000 and 13,000 tons per square inch : it appears that 
12,500 may be taken as a fair mean value. The modulus of 
rigidity C is about 5,000. 

76. Steel. The name steel is applied to a great variety of 
materials which differ mainly in the proportion of carbon they 
contain. At one end of the range is very mild steel, made in 
the Siemens furnace or in the Bessemer converter, which contains 
less than 0'2 per cent, of carbon, and differs from wrought-iron 
chiefly in the greater homogeneity which it possesses as a 
consequence of being rolled from a cast ingot instead of from a 
puddled ball interspersed with slag. At the other end of the 
range are high carbon steels, containing 0'5 per cent, or more of 
carbon, capable of being hardened and tempered by the treatment 
mentioned in § 46, and in some cases made by an entirely different 
process, namely, by adding carbon to wrought-iron in the cementa- 
tion furnace, with or without subsequent melting of the steel in 
a crucible. Mild steel containing from 0*15 to 0*25 per cent, of 
carbon has now to a very great extent superseded wrought-iron in 
engineering construction. Its tensile strength is about one-third 
greater, and its capacity for plastic yielding before fracture is also 
greater. 

Specimens of mild steel bar or plates containing about 0'2 
per cent, of carbon show in general a tensile strength of 28 to 
30 tons per square inch, and stretch about 25 per cent, on the 
8-inch length. As the percentage of carbon is increased the 



THE TESTING OF MATERIALS. 93 

plasticity diminishes, but the tensile strength becomes greater, at 
least until the percentage of carbon is as high as Oo per cent. To 
a considerable extent the strength and plasticity depend on the 
amount of work which has been done upon the ingot in rolling it 
down into the form of bar or plate, and the highest strength is 
found in wire, which is the finished product on which the largest 
amount of work has been spent. Steel wire containing a fairly 
high percentage of carbon may show a tensile strength of 80, 100, 
or even 120 tons per square inch. No rule can be laid down as 
to the relation of the strength to the percentage of carbon, for 
except in the mildest steel, the strength may be much affected 
by the state of temper, and in all cases it is also affected by 
the presence of other constituents and by the amount of rolling 
or drawing down which the ingot has undergone, as well as by 
the question whether the piece has been subsequently annealed. 
The following results obtained by Bauschinger are quoted from a 
table in Prof. Unwin's book, and will serve to give a general idea 
of the way in which the strength of steel may depend on the 
percentage of carbon it contains. The steels to which these tests 
relate were made by the Bessemer process. 



Percentage 

of 

carbon 


Tensile 
strength 
tons per 
sq. inch 


Extension in 
16 inches 
per cent. 


Shearing 
strength 
tons per 
sq. inch 


Elastic limit 

in tension 

tons per 

sq. inch 


•14 


28-1 


22 


21-7 


18 


•19 


304 


20 


23-6 


21 


•46 


33-8 


18 


22-8 


22 


•51 


35-6 


14 


25-5 


22 


•54 


35-3 


18 


25-0 


•7 9 


•55 


35-9 


18 


25-4 


21 


•57 


35-6 


18 


23-1 


21 


•66 


40-0 


14 


27-2 


24 


•78 


41-1 


11 


26-3 


24 


•80 


45-9 


9 


30-6 


25 


•87 


46-7 


s 


31-7 


27 


•96 


527 


7 


37 


31 



The contraction of area at fracture falls progressively from 4!) 
per cent, in the mildest of these steels to 10 percent, in the steel 

which is most rich in carbon. 

It is remarkable that no meat dififerenbe is found in the 



94 THE TESTING OF MATERIALS. 

modulus of elasticity whether the steel has much or little carbon. 
In the examples just quoted the modulus E was found to vary 
irregularly between 13/700 and 14.900 tons per square inch, but 
the values have no correspondence with the percentage of carbon. 

In the light of more recent experiments these values of the 
modulus of elasticity appear to be rather high. Tests made by a 
committee of the British Association (B. A. Report. 1896, p. 538), 
on two steels, one of which was much milder than the other, gave 
values of E which are lower than those quoted above and are very 
nearly the same for the milder as for the higher carbon steel. 
The following are the figures, those for E being the means obtained 
by several observers with extensometers of various types : 



Breaking 
strength 
tons per 
sq. inch 


Yield-point 

tons per 
sq. inch 


Ultimate 

extension 

or 8 inches 

per cent. 


E 
tons per 
sq. inch 


23-4 


16-0 


32 


13190 


35-6 


20-4 


24-5 


13250 



Mild steel plates have a shearing strength of 24 to 26 tons, 
and do not exhibit the same weakness in regard to shearing along 
a plane parallel to then faces which is observed in wrought-iron. 

The modulus of rigidity C was found by Bauschinger to vary 
irregularly, in the series of Bessemer steels referred to above, from 
5,320 to 5,670 tons per square inch. Like Young's modulus it 
has no obvious relation to the hardness or softness of the steel. 
In another series of tests (of Siemens steel) the mean value of 
E was 13,360, and that of C was 5,310. 

Even the mildest steel when quenched in oil or water from a 
bright red heat shows some increase of strength, with some reduction 
in ultimate elongation and raising of the elastic limit. In less mild 
steels these effects are very marked, and when the percentage of 
carbon exceeds 0"5 per cent, the process of hardening by quenching 
deprives the steel of almost all its capability of drawing out before 
rupture. 

Steel castings, while generally much stronger than iron castings, 
are less strong and decidedly less ductile than steel on which work 
has been done by forging or rolling. The tensile strength is often 
from 15 to 20 tons per square inch, sometimes as much as 25 tons 



THE TESTING OF MATERIALS. 95 

or even more. The extension before rupture is usually less than 
5 per cent. 

Many special steels are manufactured in which the iron is 
alloyed with other constituents in addition to small quantities of 
carbon and manganese. Nickel, aluminium, chromium, tungsten, 
molybdenum are among the metals used for this purpose. The 
effects of various proportions of nickel have been particularly 
studied by Mr Hadfield, who has shown that the addition of 5 or 
even 7 per cent, of that metal produces a steel which combines a 
high breaking load with much elongation and contraction of area 
at fracture*. 

* Min. Proc. Inst. C. E. vol. cxxxviii, 1899. 



CHAPTER V. 

UNIFORM AND UNIFORMLY- VARYING DISTRIBUTIONS 

OF STRESS. 



77. Use of the Stress Figure to represent a Stress 
distributed over a Surface. A stress distributed over any 
plane surface A B (fig. 45), such as an imaginary cross-section of a 




Fig. 45. 

strained piece, may be represented by setting up ordinates Aa, Bb, 
etc., from points on the surface, the length of each ordinate being 
chosen so that it represents to scale the intensity of the stress at 
the corresponding point of the surface. In this way an ideal solid 
figure is constructed, which may be called the stress figure. Its 
height exhibits the distribution of stress over the surface which 
forms its base. The volume of the stress figure represents the 
total amount of the distributed stress. A line drawn from g the 
centre of gravity of the stress figure parallel to the ordinates Aa 
etc. determines the point D through which the resultant of the 
stress on the surface AB acts. This point is called the centre of 
stress for the surface AB. 

78. Uniformly distributed Stress. When the stress is 
uniformly distributed over the surface, in other words, when its 
intensity at all points is the same, ab is a plane surface parallel to 



UNIFORM AND UNIFORMLY VARYING DISTRIBUTIONS OF STRESS. 97 

AB, and D is the centre of gravity of the surface A B. This is 
sufficiently obvious from consideration of the stress figure : it is 
also seen by taking moments about any axis YY in the plane of 




Fig. 46. 

AB. Let 8S be an element of the surface AB and let x be its 
distance from the assumed axis. Then the moment of the stress 

on the element BS is 

p$S . x, 

and the moment of the whole stress, being the sum of the moments 

for all the elements, is 

XpBS . x, 

or p^x&S, 

since in this case the intensity p is by assumption uniform. 

The resultant of the stress 

P = lpBS = pS, 

where S is the area of the surface. To produce the same moment 
this resultant must act at distance x r such that 

P . x r = p^xSS. 



Hence 






p^x&S __ 'ExSS 
= ptf 8 



which is also the equation for the distance of the centre of gravity 
of the surface from the assumed axis. 

Thus, for example, if the stressed piece is a tic-rod of uniform 
section, in which the distribution of stress is known bo be uniform, 
the resultant pull must act along the axis of the rod, namely, along 
the line which cuts each cross-section in its centre of gravity. 
For brevity we may speak of this as an axial pull. 

E. s. m. 7 



98 



UNIFORM AND UNIFORMLY VARYING 



It does not however follow that an axial pull will necessarily 
produce a uniformly distributed stress. Other distributions of 
stress which will serve to bring the resultant 
into the line of the axis can readily be imagined : 
in particular the resultant will be axial if the 
distribution is symmetrical about the axis how- 
ever much it may vary from point to point along 
radial lines. The question is one of much prac- 
tical importance, Under what conditions (if any) 
will an axially applied load produce a uniformly 
distributed stress? 




Fig. 47. 



Consider a tie-rod such as that sketched in 
fig. 47, in which there are variations of section, 
and suppose the pull to act along the axis. It is 
clear that at a section A B near the fastening no 
approach to uniformity in the distribution of the 
stress can be expected. The central part of that 
section, lying as it does directly under the pin hole, 
bears but little of the load : nearly all is borne 
by the outer portions. Again, on the section CD, 
near a place where the area suddenly changes, 
the central portion has to bear nearly all the load. 
Or again, on EF the intensity is greater near the 
edges than in the middle. But as we recede from 
these exceptional places, advancing along parts of the bar where the 
section is uniform, we find a more and more close approach to 
uniformity in the stress, and at sections such as GH or JK the 
variation is probably slight. The strains to which the variable 
stress gives rise on such sections as AB or CD tend to equalize the 
action on neighbouring layers. Thus, for example, when the 
central part of CD is more pulled than the sides the greater 
stretching of the material in the centre there produces a shear 
which makes the lines along which the pull acts spread out, in 
sections above CD, into the portions of material at the sides. 

This equalizing effect of the strain occurs, to a greatly in- 
creased degree, when the elastic limit is exceeded. In a plastic 
material especially, like mild steel or good wrought-iron, the flow 
of the material in those places where the stress first passes the 
yield point tends to relieve these of some of their excess of stress 



DISTRIBUTIONS OF STRESS. 99 

and to throw a larger proportion on other parts of the section. 
In the testing of non-plastic metal there is some difficulty in 
getting a fair test, because the inequalities of distribution which 
necessarily exist in the neighbourhood of the grips are apt to 
cause fracture to occur there ; but in the testing of plastic metal 
this difficulty does not present itself for the reason just stated, 
and fracture tends to take place where the section is most free to 
contract, namely, at or about the middle of the clear length. 

Even however for stresses lying within the elastic limit the 
equalizing effect of the strains is so considerable that for the 
purpose of engineering calculations it is generally justifiable to 
assume that an axially applied load produces a practically uniform 
distribution of stress, except at or near shoulders, nicks, holes, and 
other places where a change of section is found. It is taken for 
granted in this statement that the piece is in a state of ease 
before the load is applied. 

Subject, then, to these reservations it is usual to assume that 
'any load P applied axially to a piece whose area of section is $ 
will produce a stress the intensity of which may be taken as equal 

P . . . 

to -~ at all points of the section, to a degree of approximation 

sufficient in calculations relating to the strength of the piece. 

79. Uniformly-varying Stress. When the top of the 

stress-figure (§ 77) is a plane inclined to the plane of the surface 
on which the stress acts, the stress is described as uniformly- 
varying. The intensity of the stress is then proportional at any 
point to the distance of that point from a certain line in the plane 
of the surface, namely, the line in which the top and bottom planes 
of the stress figure meet when produced if necessary. Uniformly- 
varying stress is illustrated in fig. 48. There MN is the line in 
which the plane of the stressed surface AB is met by the upper 

LofC. 



N/ 

Mr-'-' 




Pig. 18. 



100 UNIFORM AND UNIFORMLY VARYING 

bounding plane of the stress figure. This line is called the 
Neutral Axis of the uniformly-varying stress. It lies at right 
angles to the line AB which is assumed to be the direction along 
which the intensity of stress varies most rapidly. There is no 
variation in direction parallel to the neutral axis. The intensity 
of stress p at any point may be written 

p — ax, 

where x is the distance of the point from MN and a is the rate of 
variation of the stress in the direction of the line AB : in other 
words, a is the amount by which p increases per unit of length in 
the direction of AB. 

Uniformly-varying stress is practically important because it 
occurs (for stresses within the elastic limit) in a bent beam, in a 
tie-rod when subjected to non-axial pull, and in a long strut or 
column, even when the push is originally axial, after the column 
has become bent so that the axis no longer coincides with the 
direction of the resultant thrust. The stress in beams and in 
struts will be considered in some detail later. Another example 
of uniformly- varying stress is found in a masonry pier or arch where 
the line of resultant thrust does not pass through the centre of 
gravity of the joint or section over which thrust is distributed. 

It is obvious from consideration of the stress figure that in a 
uniformly varying stress the resultant falls to one side of the 
centre of gravity of the stressed surface : in other words, the 
resultant is non-axial. And subject to qualifications similar to 
those which have been explained in dealing with uniformly 
distributed stress, the converse is approximately true for stresses 
lying within the elastic limit. That is to say, a non-axially 
applied load may be taken as giving rise to a stress which 
approximates more and more nearly to a uniformly-varying dis- 
tribution the further the section dealt with is from any place 
where the distribution is disturbed by shoulders or holes or any 
such alterations in the form of the section. Thus, for example, a 
long tie-rod of uniform section, the fastenings of which lie 
excentrically so that the pull is non-axial, will have a stress which 
to all intents and purposes is uniformly-varying except near the 
fastenings. The stress in a loaded hook is another example : there 
the resultant passes so far from the centre of the section that 



DISTRIBUTIONS OF STRESS. 



101 



while the inner edge is in tension, the outer edge is in com- 
pression. 

80. Uniformly- varying Stress forming a Couple. When 

the neutral axis MN (fig. 48) of a uniformly- varying stress lies, as 
in that figure, outside of the stressed surface, all parts of the surface 
are exposed to stress of one sign. The neutral axis may however 
lie within the surface, and it then divides the surface into two 
parts on one of which there is pull and on the other there is push. 
A particular case of much practical interest occurs when the 
neutral axis passes through the centre of gravity of the stressed 
surface. The whole amount of the pull on one side of the neutral 
axis is then equal to the whole amount of the push on the other 
side : the resultant of the stress is not a single force but a couple. 
For the quantity 1%8S has the same value over the positive region 
lying on one side of an axis through the centre of gravity as it 
has over the negative region on the other side of the same axis : 
multiply it by the constant a expressing the rate of variation of the 
stress and we have equal values of SaxSS, or LjjSS, when sum- 
mation is made separately on the positive and negative sides. 

Such a couple stress is represented graphically in fig. 49, where 




Fig. 49. 

AB is a side elevation of the plane surface on which the stress 
acts, the direction of AB being that along which the stress varies. 
The neutral axis passes at right angles to AB through C, which is 
the centre of gravity of the stressed surface. The volume of the 
wedge ACE, representing in the stress figure the negative part of 
the stress, is equal to that of the wedge BCF t which represents the 
positive part of the stress. The greatest intensity o( negative 
stress £>) occurs at .-1, and the greatest intensity of negative Stress 



102 



UNIFORM AND UNIFORMLY VARYING 



p» occurs at B. Calling x x and x 2 the distances of A and B re- 
spectively from the neutral axis through C we have 

1 2 

To find the moment M of the couple we have to find the sum of 
the moments of all the elements, in other words, to integrate 
x . pclS over the whole surface: 

M-fxpdS 

= ajx*dS = al, 

where / is the moment of inertia of the surface about the neutral 
axis through G. This may also be written 

Xi X-2 

81. Analysis of any Uniformly- varying Stress into a 
uniform stress and a couple. A stress such as that shown in 
fig. 48 or fig. 50 by the figure AabB may evidently be regarded as 




Fig. 50. 

made up of a uniformly distributed stress Aa'b'B together with 
a uniformly- varying stress whose resultant is a couple, namely, 
aa'b'b. The intensity of the uniform component, p , is the inten- 
sity which the given stress has at the centre of gravity of the 
stressed surface. The resultant is equal to p S, S being the area 



DISTRIBUTIONS OF STRESS. 103 

of the surface, and it acts at such a distance x r from C that its 
moment about C is equal to the moment of the couple component 
aa'b'b. 

Hence to find x r which is the distance of D, the centre of stress, 
from G we have 

p S .x r = al 

where a is, as before, the rate of variation of the stress, namely, 

ro — £1 or F* /o or F* — U f anc [ / i s the moment of inertia of the 

surface about an axis through G perpendicular to the direction AB 
along which the stress varies. 

The equation x r = — ~ = 7-^ — ^ — ~ 

allows the position of the resultant to be found when the stress is 
specified by giving the extreme intensities p x and p 2 . If however 
the position of the resultant is given, the extreme intensities are 
found thus : 

p Sx r = aI = (Po ~ Pl)I , 

1 

S.-i i It. &x. r x 2 \ 

lmilarly p 2 = p {) ( 1 -f — — - 1 . 

82. Extent to which Stress may be Non-axial without 
reversing its sign at the edge of the stressed surface. The 

amount by which the resultant of a uniformly-varying stress may 
deviate from the centre of the stressed surface without reversing 
the sign of the stress on any part of the surface is found by writing 

the condition for which is that Sx r x\ = I, or 



hence, p x = p 1 1 — 



Taking the particular case of a circular surfaoe (radius r) NNr 

7T?''' V 

have /= -. , S = 7rr-, and a^mri hence x r = . 

4 4 



104 



UNIFORM AND UNIFORMLY VARYING 



The stress will therefore have the same sign over a circular sur- 
face provided the resultant does not deviate from the centre by 
more than one-fourth of the radius. 

In a rectangular surface such as a joint in masonry a similar 
calculation shows that the deviation mav amount to one-sixth of 
the width of the surface without causing the stress to reverse its 
sign at the opposite edge. Hence the rule is sometimes followed 
in the design of masonry arches, of keeping the resultant thrust 
between neighbouring blocks within the middle third of the joint, 
in order that no part of the joint may be exposed to tensile 
stress. 

In a joint formed without cement, or in one where the cement 
has become ineffective in offering resistance to pull, the conse- 
quence of allowing the resultant to deviate beyond the limit of 
the middle third would simply be to put a part of the joint out of 
action. That is to say, on the off side of the joint there would 
be, over a certain area, no stress at all, and on the remainder there 
would be compression, distributed in a uniformly- varying manner. 
The case in question is illustrated in fig. 51. From A to E there 




Fig. 51. 

is no stress, the point E being found from the consideration that 
EB is three times the distance of D, the centre of stress, from C. 
In masonry piers and retaining walls it is by no means uncommon 
to find the resultant passing further from the centre than the 
middle third. 

83. Simple Bending. The stresses which are produced in 
a beam by the application of any system of loads will be con- 



DISTRIBUTIONS OF STRESS. 



105 



sidered in the next chapter, but we may notice here a specially 
.simple case in which the stress is of the kind illustrated by fig. 49. 



w n 



k-/ r H 



Fig. 52. 



w., 



k— U->>. 



F 2 



Let a beam be loaded as in fig. 52 with loads W Xi W 2 , applied at 
points whose distance from the supports are l x and Z 2 , and let 

WJ, = W 2 l 2 . 

Then the reactions at the supports, F x and F 2 , are respectively 
equal to W t and W 2 . Consider the stress in any vertical section 
A B of the beam, taken between the points of application of W 1 
and W 2 . The beam is divided by such a section into two portions a 
and /3. The only external forces acting on /3 are the couple formed 
by W 2 and F 2 , and these must be balanced by the forces which a 
exerts against /3 in consequence of the stress at the section AB. 
In other words, the stress has the character of a couple, whose 
moment is W 2 l 2 . We might have got the same result by con- 
sidering the equilibrium of the portion a. The only external 
forces acting on it are the couple made up of W x and F lt and these 
arc balanced by the forces which j3 exerts against a at the section : 
hence the stress at the section is a couple whose moment is WJ X — 
a result which is in agreement with that just arrived at, since 

WA = W 2 l. 

The moment of the stress on the section is called the Bending 
Moment. The bending moment is in this case the same for all 
sections lying between W z and W.,. 

If the stress be within the elastic limit it will be distributed 
in the uniformly-varying manner illustrated in fig. 53, with the 
neutral axis passing horizontally through the centre of gravity of 
the section. Calling //, and y t the distances of the top and bottom 



106 



UNIFORM AND UNIFORMLY VARYING 



respectively from the neutral axis, we have at the top the greatest 
intensity of compressive stress 

Pi- J 




Fig. 53. 



and at the bottom the greatest intensity of tensile stress 

My, 



P-2 = 



I 



where M is the bending moment and I is the moment of inertia 
of the section about the neutral axis. The intensity at any point, 
distant y from the neutral axis, is 

My 

P= I' 



84. Influence of Bending beyond the Elastic Limit on 
the Distribution of the Stress. The assumption made in the 
last paragraph, that a bending moment gives rise to a uniformly- 
varying distribution of stress applies only when the material is 
homogeneous and when the greatest intensity of stress falls below 
the elastic limit. Hooke's Law is supposed to be true for all parts 
of the beam. 

If, however, the bending moment be increased, non-elastic 
strain will begin as soon as either p x or jh exceeds the corre- 
sponding limit of elasticity. The distribution of the stress will 
then be modified. The outer layers of the beam are taking 



DISTRIBUTIONS OF STRESS. 



10' 



permanent set while the inner layers are still following Hooke's 
Law. As a simple instance it will suffice to consider in a general 
way the case of a material which is strictly elastic up to a certain 
limit of stress, and then so plastic that any small addition to the 
stress produces a relatively very large amount of strain — a case 
not far from being realized in good wrought-iron or mild steel. 
When a beam of such material is overstrained the diagram 
exhibiting the distribution of stress will take a form generally 
resembling that sketched in fig. 54 or fig. 55. In fig. 54 it is 




Fig. 55. 



assumed that the elastic limit is the same for tension as for 
compression, with the effect that the distribution remains sym- 
metrical about the original neutral axis. In fig. 55, on the other 
hand, it is assumed that the elastic limit is lower for compression 
than for extension, in consequence of which the neutral axis shifts 
towards the tension side when the beam becomes overstrained. 

When the overstrained beam is relieved from external load it 
is left in a state of internal stress, the general character of which 
(for the case of fig. 54) is indicated by the dotted lines in that 
figure. This internal stress satisfies the condition that its sum 
and also its moment vanish over the section as a whole. 



85. Modulus of Rupture. In consequence of the action 

which is illustrated, in a somewhat crude manner, by figs. .">4 and 
55, the bending moment M x which will break a beam cannot be 
calculated from the ultimate tensile strength f t or from the 
ultimate compressive strength f c by using fche formula 

Vi //■-■ 

because the distribution of stress assumed in finding tlii^ relation 



108 UNIFORM AND UNIFORMLY VARYING DISTRIBUTIONS OF STRESS. 

between bending moment and stress ceases to exist as soon as 
overstraining begins. 

But when experiments are made on the ultimate strength of 
bars to resist bending, it is not unusual to apply a formula of this 
form to calculate an imaginary stress f which receives the name of 
the Modulus of Transverse Rupture. Let the section be such that 
y-L = y 2 . Then the modulus of transverse rupture is denned as 



/ = 



/ 



where M 1 is the value to which the bending moment has to be 
raised in order to break the bar. 

This mode of stating the results of experiments on transverse 
strength is unsatisfactory, inasmuch as the modulus of rupture thus 
determined will vary in beams of the same material having different 
forms of section. When a plastic material in which the tensile 
and compressive strengths f t and/ c are equal is tested in the form 
of an I beam in which the top and bottom flanges form nearly the 
whole of the section, it will have a modulus of rupture not far from 
equal to f t or f c . On the other hand, if the material be tested 
in the form of a rectangular bar, the modulus of rupture may 
approach a value one and a half times as great. For in the latter 
case the distribution of stress may approach an ultimate condition 
in which the upper half of the section is in uniform tension f t and 
the lower half is in uniform compression of the same intensity. The 
moment of the stress is then equal to ifthh 2 where b is the breadth 
and h the depth of the section, while by definition of the modulus 
of rupture f we have 

Values of the modulus of transverse rupture are generally to be 
understood as referring to bars of rectangular section. 

In a material such as cast-iron, whose tensile and compressive 
strengths are very different, the modulus of rupture is found to 
differ widely from either of these strengths. Experiments on the 
cross-breaking of rectangular bars of cast-iron generally give 
values of the modulus of rupture ranging from 14 to 20 tons per 
square inch, or fully double the values which are found in tests of 
tensile strength. 



CHAPTER VI. 



STRESS IN BEAMS. 



86. Character of the Stress in Beams. In general the 
loads, as well as the supporting forces, applied to a beam act at 
right angles to the beam's length. In the special case already 
considered in § 83 the stress at any section is a bending couple 
simply : in the more general case is a bending couple together 
with a shearing stress in the plane of the section. 

Imagine a beam loaded in any manner. Let HK (fig. 56) be 



H v 



—X, 



V 



•Xo- 



F« 



B 

Fig. 56. 



any cross-section. Between the two parts A and B into which 
this section divides the beam there is a stress, the amount and 
character of which is to be found by considering the equilibrium 
of either portion A or B. The portion B, for example, is in 
equilibrium and therefore the loads and supporting force applied 
to it, namely, F u F 2 , F. M F 4 , arc balanced by the forces which A 
exerts against B in consequence of tho stair of stress which exists 
at the section HK. The system of applied forces /*',, F, etc., may 
be resolved into a single force and a single couple, by referring 



110 STRESS IN BEAMS. 

each force in turn to the plane of the section. Thus F 1 acting 
where it does act is equivalent to an equal and parallel force acting 
at HK together with a couple whose moment is equal to F x x 1} 
where x x is the distance of the force from the section. Similarly 
F 2 is equivalent to a force equal and parallel to F 2 acting at HK 
together with a couple whose moment is F 2 x 2 , and so on. Hence 
the system of applied forces as a whole is equivalent to a couple 
whose moment is 

XFx 

and to a force, in the plane of HK, and parallel to the applied 
forces, equal to 

2F. 

The former constitutes the Bending Moment at the section : the 
latter constitutes the Shearing Force. 

In other words, the stress on HK must be such as to 
equilibrate first a couple whose moment is *EFx and second a force 
IF tending to shear B from A. In these summations regard must 
of course be had to the sign of each applied force : in the case 
sketched, for example, the sign of F 4 is opposite to that of the 
other forces. 

We conclude then that the stress on any section of the beam 
may be regarded as due to a Bending Moment M equal to the sum 
of the moments (about the section) of the externally applied forces 
on one side of the section (SFx), and a shearing force equal to the 
sum of the forces on one side of the section. It is a matter of 
convenience only whether the forces on B or those on A be 
considered in reckoning the bending moment and the shearing 
force at the section which separates A from B. 

The bending moment causes (for action within the elastic 
limit) a uniformly- varying normal stress of the kind described in 
§ 83. The shearing force causes a shearing stress distributed over 
the plane of section in a manner which will be discussed later. 
This shearing stress in the plane of the section is (by § 12) 
necessarily accompanied by an equal intensity of shearing stress in 
horizontal planes parallel to the length of the beam. 

87. Stress due to Bending Moment. The stress due to 

the beading moment is the thing chiefly to be considered in 
practical problems relating to the strength of beams. It consists, 



STRESS IN BEAMS. Ill 

in an ordinary beam, of longitudinal push in filaments above the 
neutral axis and longitudinal pull in filaments below the neutral 
axis. In a cantilever, which tends to " hog " instead of " sag " 
under the action of the loads, the pull is on the upper and the 
push on the lower side : the bending moment is then opposite in 
sign to the bending moment of a beam supported at both ends and 
loaded at intermediate points. 

Whether positive or negative in sign, the bending moment 
produces (for stresses to which Hooke's Law applies) a distribution 
of stress of the kind sketched in fig. 53, with a neutral axis at the 
centre of gravity of the section. The intensity p, at any distance 
y from the neutral axis is, by § 83, 

My 

where M is the bending moment and i" is the moment of inertia 

of the section about the neutral axis. The greatest intensities of 

pull and push occur at the top and bottom edges, their values 

being 

My, My 2 

pi = —¥- anc * P* = ~ f ' • 

88. Particular Cases. One or two examples may be useful. 
Suppose the section of the beam to be a rectangle, of width b and 
depth h, and to stand with the side h vertical. Then 



bh* 

"12 ' 


h 


Pi=2V 


6M 6M 

~ bli> " Sh ' 



and 

where 8 is the area of the section. 

The advantage, in point of economy of material, which is 
gained by using a deep and narrow section is obvious. The stress 
in a rectangular beam varies inversely as h for a given bending 
moment when $ is constant. 

In a bar of square section set with its diagonals horizontal and 

vertical, the moment of inertia / has the same value as if the 

b 4 
sides were horizontal and vertical, namely, - where b is the 



112 STRESS IN BEAMS. 

length of each side. Hence^ for a given bending moment is greater 
in the ratio in which y 1 is greater, namely, in the ratio V2 : 1, 

6V2.M 



P1-P2 



Sb 



7TC/ 4 

In a soiid circular section of diameter d, I = -^-r an d 

o-± 

_S2M_8M 

If the section is a hollow circle, like that of a bicycle tube, 

in which the thickness is small compared with the diameter, / 

Sd 2 
approaches the limiting value -^- , and in the limit, when the 

thickness is relatively indefinitely small, 

P1-P2- Sd , 

89. Beam with Flanges and Web. A more advantageous 
disposal of the material is arrived at, in respect of bending strength, 
when it is concentrated at the places where the stress is greatest., 
namely, at the top and bottom. Hence in the most usual form a 
beam consists of two flanges held apart by a thin iveb or by bracing- 
equivalent to a web. The function of the web is, as will be shown 
later, to take the shearing force. The bending moment is borne 
mainly by the flanges, one of which is in tension and the other in 
compression ; and if the depth of each flange is small in comparison 
with the depth of the beam, the intensity of the stress is nearly 
uniform over the whole of each flange, at any section. 

Girders of this kind are formed by rolling solid metal with an 
I section, or are built up of plates, or are made by combining 
separate members to form frames. In some cases the beam takes 
a box-shaped section through the use of two webs instead of one. 
Often the strength to resist bending moment is calculated with 
reference to the flanges alone, the (generally small) addition to 
the bending strength which the web affords being left out of 
account. 

In that case the relation of the bending moment to the stress 
in the flanges may be expressed simply as follows. Let S lt S 2 be 
the area of the flanges, p 1} p 2 the intensities of stress on them, and 
h the height reckoned from the middle of one flange area to the 



STRESS IN BEAMS. 



113 



middle of the other. Then, neglecting the small variation of p 
over each flange, 

M =p 1 8 1 h = p 2 8 2 h, 
and consequently 

M_ M 

Pl ~ 8 1 h > P *~~SJi' 

It is clear that the greatest economy of material can be secured 
only when the sectional areas of the tension and compression 
flanges are made inversely proportional to the tensile and com- 
pression strength, so that 8 t f t may be equal to 8 c f c . In rolled 
beams of wrought-iron and steel and also in plate beams of these 
metals both flanges are usually of the same sectional area, but in a 
cast-iron beam a section such as that shown in ne. 57 would be 




Fig. 57. 

suitable, with a relatively large tension flange, the ratio of the two 
strengths being fully six to one. In a cast-iron beam the web is 
necessarily of considerable thickness and cannot properly be left out 
of account in reckoning the bending strength. 



90. Variation of Bending Moment and Shearing Force 
from point to point along a Beam. Diagrams of Bending 
Moment and Shearing Force. The bending moment and the 

shearing force in general vary from point to point along a beam, and 
they are conveniently shown by setting up ordinates the lengths 
of which represent the values of these quantities, fco any convenient 
Scale, on a base line representing the length of the beam. A few 

examples of such diagrams may be given, and the student will find 

it a useful exercise to draw others for himself. 

E. S. M. > 



114 STRESS IN BEAMS. 

1. Single load TT at the centre of the span : — 



ov 



i 




Fig. 58. 



.w 
2 



Yir. 59. 



Calling P and Q the reactions at the ends,, we have P=Q=^W. 

Let L be the span and x the distance of any section from the 
end P. then, for the moment at a section between P and TT, 

J/, = P.': 
aud for the shearing force 

For a section between TT and Q the bending moment is 
Q I L — ;/.). and the shearing force is equal to Q or %W. 

The maximum bending" moment is at the centre and its value 



is 



PL WL 



or 



The diagrams of bending moment and shearing force are 

sketched in figs. 58 and 59 respectively. We shall distinguish a 

shearing force as positive when it tends to shear the right-hand 

portion of the beam up. With this convention the shearing force 

is positive on the right-hand half of the beam,, and negative on the 

TT W 

left-hand half. It changes from + -^- to — at the place where 

the load is applied. 



STRESS IN BEAMS. 



115 



This abrupt change of the shearing force at the place where 
the load is applied must not be misunderstood to mean that there 
can be two different values of the shearing force at a single section. 
There is only one value at each section, for any given distribution 
of load. The apparently anomalous state of things at the section 
under the load is due to the conventional assumption that the 
load is applied at a point. Any real load, however much concen- 
trated, would be distributed over some finite length of the beam, 
and the change from positive to negative shearing force would be 
gradual over that length. 

2. Single load W placed at any distance c from one end : — 



I 1 




w 




Fig. 60. 




Fig. 61. 



Here Q = T and r = T 



The bending moment 



M r = P 



X 



as before, so long as x is less than c : the greatest value La reached 

, , . Wc(L-c) m , / ,. v , ,. 

when x = c and is —^= -. The (negative) shearing force id 

the left-hand portion is equal to P, and the positive shearing force 
in the right-hand portion is equal to Q. The two rectangles 
which make up the diagram of shearing force have equal arras. 

- -2 



116 STRESS IN BEAMS. 

3. Two or more separate loads : — 




w 



w a ) w 



?-. 



Q. 




Fig. 62. 









w 3 






w 2 














w x 


Fig. 63. 







Q 



The diagram in this case may readily be drawn by drawing the 
diagrams for each load considered alone, in the first instance, and 
then combining them by adding the ordinates. The separate 
diagrams are shown in fig. 62 in fine lines, and the final 
diagram, derived from them, is shown in bolder lines. The shear- 
ing force diagram may be formed by superposition in the same 
way (fig. 63). 

4. Continuous distribution of load, uniform per foot-run of 
the span : — 

Let the uniform load be w per foot-run. The reaction at each 
pier is \%vL. At any distance x from the left-hand end the bending 
moment 

M x = Px — wx . - = ^ (Lx — x 2 ). 



STRESS IN BEAMS. 



117 



This is a maximum when x = \L, its value then being 

wL 2 

The curve of bending moments is a parabola. (Fig. 64.) 




Fig. 64. 




Fig. 65. 



The shearing force 

F x = wx 



P = W Ix — ,y 

ivL 



decreases uniformly from the value — at the left-hand end, be- 
comes zero at the middle, and increases uniformly to ■=- at the 
other end. (Fig. 65.) 

5. Beam carrying a uniformly distributed load over a part of 
its length only : — 

Here the portions which are clear of the load are affected just 
as they would be if the load were concentrated at its own centre of 
gravity G. The straight lines pa and qb of the bending moment 
diagram, fig. 6Q, would meet, if produced, above G, and the curve 
from a to b is drawn by erecting on the base ab the parabola which 
would be the bending moment diagram of a beam AB carrying the 
same distributed load. 

The straight line diagram pabq represents the bending moment 
which would exist if a pair of equal loads, fcogel her equal to I hedistri- 



118 



STRESS IX BEAMS. 



buted load, were applied at A and B respectively. But in addition 
to the moment so produced, the effect of the distribution between 
A and B is to produce in that portion of the beam a supplementary 



<k 



Q 




p A G B q 

Fig. 66. 




Fig. 67. 



amount of bending, equal to that which the same load would 
produce if distributed over a beam AB resting on supports at A 
and B. This may be shown analytically, but it will be obvious 
from considerations to be brought forward in § 92 below. 

The diagram of shearing force for the same case is sketched in 
fig. 67. 

6. Beam projecting as a cantilever beyond one of its supports 
and loaded as sketched : — 

The bending moment diagram afcd, fig. 68, is obtained by drawing 
acd which is the diagram due to the load at D alone, and abe which 
is due to the load at B alone and superposing them. Or, more 
directly, we obtain afc by setting up on the sloping base ac the 
ordinates of the diagram due to the load at B. To draw the 
shearing force diagram, fig. 69, we may calculate the reaction at C. 
which is the step by which the shearing force changes there, and 
then draw the diagram from left to right- : or alternatively, sketch 
the diagram for the load at D alone, the reaction at C due to 



STRESS IX BEAMS. 



119 



AD 



that load being W D . ~jj i ; then superpose the diagram for the load 
applied at B. 




Fig. 68. 




Fig. 69. 
7. Beam with distributed load and projecting end carrying a 
single load : — 



i 



^-~ — , . : ,'"■■::'■;' 



V 



cr~ ••-. 






Fig. 70. 





Pig. 71 



120 



STEESS IX BEAMS. 



The bending moment diagram for the single load is first drawn 
abc (fig. 70) and then that for the distributed load is drawn on the 
sloping base be. Similarly for the shearing force, draw defgh for 
the single load and superpose on it the diagram gjkh for the distri- 
buted load. 

91. Graphic method of finding Bending Moments. The 

following more purely graphic method of determining the bending 
moments on a beam loaded in any manner is occasionally useful, 
Given a beam AB carrying loads W l} W 2 etc. at distances a^, a 2 etc. 
from the support A. On an ordinate at A set off the distance 
AC = QL, L being the span, and join CB. On the line CA mark 




off CD = W 1 a 1 , DE = W 2 a 2> EF '= W 3 a s and FG = W,a 4 . The re- 
maining distance GA will be equal to W- a 3 since %Wa= QL, there 
being no bending moment at the pier A. Join D with H the 
point where CB meets the line of W 1: D with / and so on. The 
line BHIJKLA is the diagram of bending moments : its height 
evidently represents the quantity 2ifa, summation being made 
in respect of the forces which lie to the right-hand side of any 
section. 



92. Relation between the Bending Moment Diagram 
and the Funicular Polygon for the same system of Loads. 

The polygon formed by a hanging cord, under any system of loads, 



STRESS IN BEAMS. 



121 



is a diagram of bending moments for a beam similarly loaded. To 
prove this consider any section of the cord such as C (fig. 73). 
The stress there may be resolved into two components, one along 





Fig. 74. 

the vertical line CD and the other parallel to the line AB, which 
represents the span of the corresponding beam. The component 
parallel to AB is the same wherever the section C be taken, as is 
evident from consideration of the reciprocal figure. Call it H. 
The force exerted on the cord at B may also be resolved into a 
vertical part and a part along DB. Then the equilibrium of the 
whole right-hand portion of the cord, from C to B, requires that the 
moments taken about D shall balance, hence 

H.CD = 2Fx 

where x is measured horizontally from the section, the sum being 
taken to the right of the section and including the vertical com- 
ponent of the force at B, which is the same as the reaction at the 
pier in the similarly loaded beam. Hence 

2Fa> 



CD = 



H 



and since H is constant CD is proportional to 2Fj\ which is the 
bending moment on the similarly loaded beam. In this proof it is 

not necessary that the line AB be horizontal. 

It is interesting to apply this proposition to such a case as the 
example do. 5 of § 90. 



122 



STRESS IN BEAMS. 



The continuous load from A to B (fig. 75) causes that portion of 
the funicular polygon to take a curved form, namely the form which 




Fig. 



(o. 



a chain whose weight represents the distributed load would take 
if hung from the points A and B. The straight line polygon 
PABQ is the one which would be got by referring the load to its 
two extremities A and B. Hence the bending moment diagram 
is properly drawn (as in § 90) by drawing the straight line diagram 
PABQ first and superposing on i5 a diagram representing the 
bending moment which the distributed load would produce on a 
beam of the span A B. 

93. Connection between Bending Moment and Shear- 
ing Force. 

Consider two sections of a beam closely adjoining one another, 
separated by an indefinitely small distance &%. Call the bending 
moment on one 21, and on the other 21 + B2I. 

We have 

2I = XFx 

the distance of each force from the second section being greater by 
the amount hx. 

Hence M + 821 = IFx + 8xZF 

and m=8x%F 

dM 



or 



2F = 



da 



That is to say, the shearing force is equal to the rate of change of 
the bending moment, from point to point along the beam. 



STRESS IN BEAMS. 123 

Hence also, in diagrams of bending moment and shearing force, 
the height of the ordinate in the shearing force diagram measures 
the gradient of the curve in the bending moment diagram, and the 
area enclosed by the curve in the shearing force diagram, between 
any two points of the span, measures the difference between the 
bending moments at these two points. 

If, for instance, we take two points on the beam where the 
bending moments are equal, the shearing force line between these 
points must enclose equal positive and negative areas. A par- 
ticular case is when there is no bending moment at each of the 
two points. This applies, for example, to the whole length of an 
ordinary beam resting on two end supports, for at each support 
the bending moment is zero. It also applies to the whole 
length of an overhanging beam such as that of example no. 6 in 
§ 90. The case of simple bending, without shearing, discussed in 
§ 83, is found only when the bending moment is uniform. Illustra- 
tions of the relations between bending moment and shearing force 
diagrams will be found in the examples which have been already 
given. 

94. Bending Moment and Shearing Force due to 
Moving Loads. The student should find it easy to establish 
the following propositions with respect to the action of moving 
loads on a beam supported at its ends. 

The bending moment at any section due to a single moving 

load is greatest when the load is at the section. Its value is 

W x (L — x) 

i L where x is the distance from one end. The diagram 

of maximum bending moments is consequently a parabola ; its 

W L 

height at the middle is - . 

The shearing force at any section due to a single moving load 

is positive when the load is approaching the section from the left, 

and negative when the load has passed the section. The greatest 

positive and negative values are found when the load is indefi- 

11' 
nitely near to the section, on each side. Their values are ' 

— W(L-x) 
and ' respectively. The diagram of maximum shearing 

Li 

forces (positive and negative) is sketched in fig. 76. 



124 



STRESS IN BEAMS. 



The bending moment at any section due to a uniform advancing 
load is greatest when the beam is wholly covered. 







Fig. 76. 

The shearing force at any section due to a uniform advancing 
load has its greatest positive value when the load covers the portion 
of the beam lying to the left of the section, and its greatest negative 
value when the load covers the portion of the beam lying to the 
right of the section. The diagram of maximum positive and 
negative shearing force is sketched in fig. 77. The values at any 

IfJOf? 1U ( ±j Xi 

section distant x from the left-hand end are 7 r^ F and Vr • 

2L 2L 




Fig. 77 



95. Distribution of Shearing Stress over the Section of 
a Beam. The shearing stress at any point in a vertical section of a 
beam is (by § 12) associated with a shearing stress of equal intensity in 
a horizontal plane through that point. If for instance AB and A'B' 
are two closely neighbouring vertical sections separated by a short 
distance Sx, the intensity of shearing stress in the section AB at 
the point H is the same as the intensity of shearing stress in the 
plane of HJ. We find the shearing stress in HJ by considering the 
equilibrium of the piece AH J A', The normal stresses on AH and 
A 'J due to the bending moments M and M + 8M respectively differ 



STRESS IX BEAMS. 



12- 



by an amount represented in the figure by the shaded diagram 
A'JKL, ACD being the stress figure for AC and A' CD' the 




stress figure for A'Cf, and A'C'L being the difference between 
them. The excess of horizontal force on one side of the piece 
AIIJA' is balanced by shearing stress on the surface HJ, and the 
whole amount of that stress is consequently equal to the total 
stress represented by the shaded figure A'JKL. 

Calling q the intensity of the shearing stress at H and f the 
width of the beam there, we have 

q£$x 

for the whole shearing stress on HJ. 

The intensity of the normal stress due to BM, at any height y 

from the neutral axis is 

yBM 

1 / ' 



and hence the whole horizontal force represented by the shaded 

yBM 



figure is 



/ 



zdy, 



where z is the width of the beam at the height //. integration 
being performed between the limits y = CA and y — CH. 

This may be written 

BMC j BM 

y-Jyzdy or / Sy^ 



126 STRESS IN BEAMS. 

where S is the area of that part of the section which extends from 
A to H, and y is the height of the centre of gravity of that part 
of the section, above the neutral axis. Equating these two ex- 
pressions for the whole horizontal force on HJ we have 

from which 

8M % _ FSy^ 

q hx' £1 & ' 

where F stands for the whole shearing force at the section con- 
sidered, the shearing force being (by § 93) equal to -j— . 

It follows that the intensity of shearing stress is in all cases 
greatest at the neutral axis, and diminishes to zero at the top and 
bottom of the section. In the particular case of a rectangular 
section, the greatest intensity of shearing stress is 

or one and a half times the mean intensity over the whole section. 
Similarly in a circular section the maximum intensity of shearing 
stress is | of the mean. 

In an I beam with wide flanges and a thin web the above 
expression for q shows that the intensity of shearing stress is 
nearly uniform over the web, and is much greater there than in 
the flanges in consequence of the much smaller value of the width 
J. A substantially accurate result is got, in such a case, by taking 
the web to bear the whole shearing force, with practically uniform 
distribution over the section of the web. 

The same intensity of shearing stress occurs in horizontal 
planes in the web, and has to be reckoned with in designing the 
rivets or other fastenings by which the web is attached to the 
flanges. 

96. Principal Stresses in a Beam. The foregoing analysis 
of the stresses in a beam, which resolves them into longitudinal 
pull and push, due to bending moment, along with shear in 
longitudinal and transverse planes, is generally sufficient in the 



STRESS IX BEAMS. 



127 



treatment of practical cases. If, however, it is desired to find the 
direction and greatest intensity of stress at any point in a beam, 
the planes of principal stress passing through the point have to be 
found. This is a particular case of the general problem of finding 
the principal stresses when the stresses in 
certain directions are known. In this case 
the problem is exceptionally simple, from the 
fact that the stresses on two planes at right 
angles are known, and the stress on one of 
these planes is wholly tangential. Let AC 
(fig. 79) be an indefinitely small portion of 
the horizontal section of a beam, on which 
there is only shearing stress, and let AB be 
an indefinitely small portion of the vertical 
section at the same place, on which there is 
shearing and normal stress. Let q be the 
intensity of the shearing stress, which is the 
same on AB and AC, and let p be the in- 
tensity of normal stress on AB: it is required 
to find a third plane BC, such that the stress on it is wholly 
normal, and to find r, the intensity of that stress. Let 6 be the 
angle (to be determined) which BC makes with AB. Then the 
equilibrium of the triangular wedge ABC requires that 

rBC cos 6 = p.AB + q. AC, and rBCsm6 = q.AB: 

(r — p) cos = q sin 6, and r sin 6 = q cos 6. 




Fig. 79. 



or 



Hence, q 2 = r (r —p), 

tan 2(9 = 2q/p, 

The positive value of r is the greater principal stress, and is of the 
same sign as p. The negative value is the lesser principal stress, 
which occurs on a plane at right angles to the former. The 
equation for 6 gives two values corresponding to the two planes »>(' 
principal stress. The greatest intensity of shearing Btress occurs 
on the pair of planes inclined at 45° to the pianos of principal 
stress, and its value is s/q 1 + £/r (by § 10). 

The above determination of r, the greatest intensity of Btress 
due to the combined effect of simple bending and shearing, Is of 



128 STRESS IN BEAMS. 

some practical importance in the case of the web of an I beam. 
We have seen that the web takes practically the whole shearing 
force, distributed over it with a nearly uniform intensity q. If 
there were no normal stress on a vertical section of the web, the 
shearing stress q would give rise to two equal principal stresses, of 
pull and push, each equal to q, in directions inclined at 45° to the 
section. But the web has further to suffer normal stress due to 
bending, the intensity of which at points near the flanges approxi- 
mates to the intensity in the flanges themselves. Hence in these 
regions the greater principal stress is increased, often by a con- 
siderable amount, which may easily be calculated from the 
foregoing formula. What makes this specially important is the 
fact that one of the principal stresses is a stress of compression, 
which tends to make the web yield by buckling, and must be 
guarded against by a suitable stiffening of the web. 



CHAPTER VII. 



DEFLECTION OF BEAMS : CONTINUOUS BEAMS. 



97. Curvature due to Bending Moment. We have to 
consider, in the first instance, the strain produced in beams by the 
action of the bending moment. The bending moment causes 
longitudinal strains, of extension on one side of the surface con- 
taining the neutral axes and compression on the other side. The 
beam, if originally straight, consequently becomes curved. In 
dealing with the curvature and deflection of beams we shall 
assume that the strains lie within the elastic limit and, as is 
always the case in practice, that the beam is stiff enough to keep 
the deflection small, and we shall in the first place exclude the 
case of a beam whose width is much greater than its depth. 

The strain on any imaginary filament taken along the length 
of the beam is sensibly the same as if that 
filament were directly compressed or extended 
by itself. Since the stress at any section 
varies directly as the height y above or below 
the neutral axis, the strain also varies directly 
as that height. Hence two plane cross- 
sections, taken near together, which are pa- 
rallel before straining become inclined to one 
another when the beam is strained, but remain 
plane. Let I (fig. 80) be the original distance 
between the two sections. At the level of the 
neutral axis this distance remains unaltered 
by the strain. At any height y above or 
below the neutral axis it chancres by the 
amount hi. By Hooke's law 

ti_ p 
I " A" 




Fi^. 80. 



E. S. M. 



130 



DEFLECTION OF BEAMS : CONTINUOUS BEAMS. 



where p is the mean stress, between the two sections, at the 
height y above the neutral axis. Further hi \ y = 1 \ R, where R is 
the distance from the neutral axis to the axis where the two 
planes of section meet. 



Hence 



P 



Suppose the planes of section are taken indefinitely near 

together, so that the bending moment M is the same for both, 

R is then the radius of curvature of the bent beam at the place 

My . . 

considered. Since p — -— ^Q may express R also in the form 

7? EI 



98. Condition of Uniform Curvature. It follows that a 
beam, originally straight, will bend into a circular arc if V (or- 
is constant. Practical cases in which this occurs are found 
(1) when a beam of uniform section is exposed to a uniform 
bending moment, and (2) when a beam is so designed that its 
depth and flange stress are both 
uniform. Calling the dip in the 
middle iij, we have in that case 
(fig. 81), 

CE . CD = CA . CB, 



from which, since u z is small 

D _ DM 
Ul ~SR~8EI- 

The greatest slope is at each end, 
and is 

h ~2R~2EI' 

R being by assumption large compared with L. 




FiR. 81. 



DEFLECTION OF BEAMS : CONTINUOUS BEAMS. 



131 



99. Relation of Curvature, Slope, and Deflection. 

Denoting distance measured along the span by x, slope by i, 
and deflection by u, we have in all cases where 
the deflection is very small, so that 8x may be 
taken as sensibly equal to 8s (fig. 82), 



du _ . 

8% 



8x 
R 



= 8i 



Then the curvature 

1 _ di d 2 u 
R dx dx 2 ' 
the slope 

J -tt 

and the deflection, measured from below up- 
wards, 

u — Jidx. 

These equations allow us to find the slope and 
the deflection when R cam be expressed as a 
function of x. The following are examples. 




Fig. 82. 



100. Examples of Slope and Deflection in Beams and 
Cantilevers. 

(1) Beam of uniform section with uniform bending moment. 

1 M 
R~ EI* 

._ M _^M f _ Mx 
l ~JR dx ~EIJ EI' 

If we take the origin at the middle of the beam the constant of 
integration is zero, since i = when x = 0, and the greatest slope 

is found by writing x = -^ , namely 

. M L 
h ~lEI2 i 

which agrees with the result got geometrically in the last 
paragraph. 

Similarly the deflection 

f. 7 [Mx . M f , Ma? 
u = I idx = £j dx= -]fj J tod® - 2EI ' 

9- 2 



132 DEFLECTION OF BEAMS: CONTINUOUS BEAMS. 

and its greatest value, namely the rise of the ends of the beam 

above the middle, is 

ML 2 

Ul ~ 8EI 
as before. 

(2) Beam of uniform section with a single load W at the 
centre of the span. 

Taking the origin at the centre as before, in order to make the 
constants of integration vanish, we have 

W/L 
2 

w r/L 



At the ends, 



i, = 



WL 



To find the deflection, 

W 

u = Jidx = ^pj f(Lx — x 2 ) dx 

Wx 2 (L _ x\ 
~ Wl \2 " 3/ " 
At the ends, 

W1J 

Ul ~ 4>8Er 

(3) Beam of uniform section with a uniformly distributed 

load. 

Again, taking the origin at the centre, 

M _ wL 2 wx 2 
- ~g w > 

i=\-dx = ^ I \[--x 2 



wx (L 2 x 2 



At the ends 



2JEI V 4 
wL 3 



h UEI' 



To find the deflection, 

,. 7 w f/L 2 x x 3 \ 7 

wx 2 (L 2 x 2 \ 
~ SEI \2 " 3 J ' 



DEFLECTION OF BEAMS: CONTINUOUS BEAMS. 133 

At the ends owL 4 



384&T 

Corresponding results for loaded cantilevers are readily found 
in the same way. 

Both in beams supported at the ends and in cantilevers the 
greatest slope and greatest deflection may conveniently be ex- 
pressed in the form 

. _ WD , WD 

where W is the total load, distributed or not, and n and n' are 
factors depending on the uniformity or non- uniformity of the section 
and on the mode of loading. The following table gives numerical 
values of n and ri in various cases where the section is uniform. 
L stands for the total length of the beam or the cantilever. 

Beam of uniform section with single load 
at centre 

Beam of uniform section uniformly loaded 

Cantilever of uniform section with single 
load at end 

Cantilever of uniform section uniformly 
loaded 

Similar expressions will apply in the case of beams of uniform 
depth and uniform strength (uniform flange stress) if Ave under- 
stand / to refer to the central section in the case of a beam, or to 
the section at the fixed end in the case of a cantilever. The 
curvature is, as we have seen, uniform, and the factors n and n 
take the following values. 

Beam of uniform strength and depth, with 
single load at centre 

Beam of uniform strength and depth uni- 
formly loaded 

Cantilever of uniform strength and depth, 
with single load at end 

Cantilever of uniform strength and depth 
uniformly loaded 

The deflection due to a combination of Loads may be found by 
summing the deflections clue to the loads considered separately. 



n 


n 


1 


1 


16 


48 


1 


5 


24 


384 


1 


1 


2 


3 


1 


1 


6 


8 



n 


n' 


1 


1 


8 


32 


1 


1 


16 


64 


1 


1 
2 


1 


1 


2 


4 



134 



DEFLECTION OF BEAMS : CONTINUOUS BEAMS. 



101. Deflection of a uniform beam under a single load 
placed anywhere. As a further example of the general method 
we may take the case of a beam of uniform section with a single 
load W placed at a distance a from the end P and b from the 




Fig. 83. 

end Q. Take the place where the weight is applied as origin 
and consider first the portion of the beam which lies to the 
right. At any point in it the bending moment is Q (b — x). The 
change of slope, in going from the origin towards the right is 

f If O 

I ^j-dx or -fTf$Q> — x) dx, and the whole slope at any point on the 

right is 

• • Q C x ,r 

l=zl ° + ~EI (P- x )doc, 

where i is the slope at 0, 

. . Q /. x 2 
l = l ° + El{ h0 °-2 

The deflection, measured up from the horizontal line through 0, is 

u = i x + jr T I ('bx - -J dx. 

Hence the height of the end Q above the horizontal line 
through 0, namely cd + de in the figure, where the deflection of 
the beam is, of course, excessively exaggerated, is 

Q b* 



i b + 



EI' S' 



The line gd is drawn through in the direction of the slope of 
the beam at 0. 

Similarly the height of the end P above the horizontal line 
through (namely gh —fg in the figure) is 

P a 3 . 



EI'S 



— i a a. 



DEFLECTION OF BEAMS : CONTINUOUS BEAMS. 



135 



Equating these two expressions for the deflection of below 
the level of the ends of the beam we have 



io(a + b) = Ml (Pa s -Q¥), 
and, substituting the value of i found thus, the deflection at is 

1(0 SElV a a+b 

_ Pa s b + Qab s 
~8EI(a + b)' 

Wa 2 b 2 
" SEI (a + 6) * 

A graphic method of solving the same problem will be found 
in 8 104. 



102. Transverse Bending. Anticlastic Curvature. As- 
sociated with the longitudinal bending of beams is a transverse 
bending with opposite curvature. This results from the lateral 
contraction of the longitudinally extended filaments and the 
lateral expansion of the longitudinally compressed filaments. An 
originally rectangular section tends to take a form like that- 
sketched in fig. 84, the beam being one supported at its ends, 
so that the bending moment produces longitudinal 
extension above the neutral axis and longitudinal 

compression below it. The lateral strain being - of 
the longitudinal strain, the anticlastic or transverse 
curvature to which it gives rise is - of the longi- 
tudinal curvature, and the radius of transverse 
curvature is Ra. 




Figj 84 



This however assumes that each filament is perfectly free to 
expand or contract laterally, a thing which is never more than 
approximately true and becomes loss true the greater Is the 
width of the beam. When a wide flat strip is bent as a beam it 
necessarily remains nearly flat: in other words, the horizontal 
lateral strain which would give rise to transverse curvature is to a 



136 DEFLECTION OF BEAMS: CONTINUOUS BEAMS. 

great extent prevented. As a consequence, the strip is stiffer in 
regard to longitudinal bending than it would be if each filament 
were free to take lateral strain. The case approximates to that 
considered in § 30, namely where lateral strain is free to take 
place in one direction (namely vertically), but is prevented from 
taking place in the other direction (horizontally). The ap- 
propriate modulus for the longitudinal strain is then 

Ea°- 



c^-r 



and the radius of longitudinal curvature, instead of having the 

value 

EI 

M' 

as it has (very nearly) when the transverse dimensions of the 
section are small, has a value approximating to 

Ela 2 



M(a*-1)' 



The ordinary theory of bending applies only when the section 
is so comparatively narrow that the anticlastic bending due to 
lateral strain is substantially free to take place, and this holds good 
in most actual beams. The transverse flexure is not in general 
of practical importance. 

103. Resilience of a Beam. The work done in bending any 

short portion Soc of a beam is — — where M is the bending moment 

and Bi is the amount by which the slope changes from one to the 
other end of the element of length 8x. Hence the whole work 
done in bending the beam is 

U=ifMdi, 

integration being performed from end to end. 

1 EI 

Since 8i = -== &% and R = -^ we may write 
R M J 

[M 2 
The following are particular cases : (1) Beam of uniform section 



DEFLECTION OF BEAMS : CONTINUOUS BEAMS. 137 

subjected to a uniform bending moment, and therefore assuming 
uniform curvature, 

2EI 

Since -=■ = - this may be written 
I y J 

2Ey* 

And when the section is rectangular this becomes 

TT PihbL 
6E ' 

p l being the intensity of stress at the top or bottom. Thus the 
resilience of a rectangular bar when uniformly bent is 

Pi 
6E 

per unit of volume, or one-third as great as the resilience of a 
piece uniformly stressed by a simple pull or push of the same 
greatest intensity p x . 

This result might have been reached at once from the 
consideration that in the beam the stress varies uniformly across 
each half of the section from zero to p 1 and consequently the mean 
value of p 2 is one-third of the extreme value p*. 

(2) Beam of uniform section with a single load W at the 
centre. 

Take the origin at one end and calculate the resilience of one- 
half the beam : 



2 ~ 2EI J M ax " 2JSIJ V 2 ) 8EI 



a-' 

3 



Hence, writing x = — , we have for half the beam 

2~192EI' 

W*D 

and the resilience of the whole beam is ^,. 7 , r . 

90 El 

This might also have been got as half the product of the load 

II7/ ; 
by the deflection m 1} which is . '. . 
J 48E1 



138 DEFLECTION OF BEAMS: CONTINUOUS BEAMS. 

Expressed in terms of the greatest stress at the middle section 
the above expression for U becomes 

Pi'IL 

§E y r 

In a rectangular beam loaded at the middle this gives 

ISE ' 

making the mean resilience per unit of volume equal to 

PL 

18^' 

or one-ninth of the resilience of a piece uniformly stressed with 
the intensity p 1 . 

This again is a conclusion which might be arrived at by 
considering that the mean value of p 2 across any section is ^p x 2 
for that section, and that the mean value of p x 2 along the beam is 
one-third of the value at the middle section. Hence the mean 
value of p 2 for the whole volume of the beam is one-ninth of the 
value of p* at the middle section. 

104. Graphic method in the treatment of Deflection. 

The three quantities 

Curvature, -^ , 

Slope, i, 
Deflection, u, 

are related to one another in the same way as the three quantities 

Load per foot run iv, 
Shearing force, F, 
Bending moment, M. 

-n du , di 1 . ., dM „ , dF 

r or -=- = i and -=- = ^, while -^— = h and -±- = w. 
ax ax K ax ax 

Hence, if we assume an imaginary load iu' equal to the 

1 if 

curvature ■=, or ^^., the shearing force which w would cause 
R EI ° 

measures the slope due to the real load, and the bending moment 

which w' would cause measures the deflection due to the real load. 



DEFLECTION OF BEAMS : CONTINUOUS BEAMS. 



139 



The problem therefore resolves itself into finding the shearing- 
force and bending moment which would be produced by the 
imaginary load w ', and in practice this is in general best done by 
drawing the diagram of shearing force and bending moment for 
the imaginary load. 

The method may be illustrated by an example. 

In a beam of uniform section with a single load W at the 

M 

middle, the imaginary load equal to the curvature is w' = ^ . 

WL 

This has its greatest value at the middle, namely Tpj- Its 

mean value is half this, and hence the reaction at each pier due to 

WL 2 
the imaginary load w' is YFWf 

The diagram of w' is sketched in fig. 85, and that of the 
shearing force due to w' in fig. 86. The greatest slope in the 




16EI 



Fig. 85. 




Fig. 86 




Fig. 87. 



actual beam is the greatest value of the shearing force due !■> 

WD 

w' and is therefore , .. „ T i as was found otherwise m § 100. 



140 



DEFLECTION OF BEAMS: CONTINUOUS BEAMS. 



The diagram of bending moment due to w is sketched in 
fig. 87. At the middle of the span the value is 

WD L_ WD^ L_ WD 
WEI' 2 16EI' 6 ~ 18EI' 

This measures the deflection at the centre produced by the actual 
load, and is in agreement with the result found in § 100. 

As another example we may take the case dealt with in § 101, 
of a beam of uniform section carrying a single load at any point.. 
The diagram of imaginary load w is sketched in fig. 88. Let 




Fig. 88. 

P' } Q' be the reactions due to it, and F a , F h be the total amounts 
of the imaginary load distributed over the portions a and b respec- 
tively. Then calling the reactions at the support of the actual 
beam P and Q as before, we have 



, ^ x Pa Qb 

greatest tu (at U) — -^j = -wr = 



Wab 



EI EI EI (a + by 



a 2EI' 



F h = 



2EI' 



_ F a .ia + F b (a + ib ) iPa* + lQb*(a + ib) 
V ~ a+b EI(a + b) 

The deflection in the actual beam at is the moment at due 
to the imaginary load w\ namely 

Q'b-F b ^b. 

Substituting the values given above for Q' and F b this becomes 

Pa 3 b + Qa¥ Wa?b 2 



SEI (a + b) ' SEI (a -f b) 



as in 



101. 



DEFLECTION OF F.EAMS : CONTINUOUS BEAMS. 141 

105. Additional Deflection due to Shearing. The 

shearing strain in a beam produces a supplementary deflection 

which is in general too small to be of practical account. We 

may examine its value in a particular case, namely that of a beam 

of uniform rectangular section, loaded with a weight W at the 

middle. The total shearing force is in this case uniform in 

W 
amount along the whole length, its value being — and it is 

distributed in the same manner at each cross section. 



By § 95 its intensity q at any height y from the neutral axis is 

6 v\ 



airg-,) 



The work done in producing the shearing strain is by § 22 
~~ P er un it °f volume at any place. The work done in shearing- 



is therefore 



2C hdy 



per unit of length of the beam. Hence the whole work done in 
shearing the beam is 

On integrating from y = ^ to 2/ = — ~ this gives 

3 W*-L 

s ~20hbC' 

The work done in bending the beam (apart from shearing) is 
by § 100, 

{Jb "Mei' 

The total deflection at the middle, where the load is applied, muel 
be such as to give this quantity when multiplied by half the Load, 
Hence the deflection is 

:\\YL 117. 

10/<6(' + 4NA7 



142 DEFLECTIOX OF BEAMS: CONTINUOUS BEAMS. 

The second of these terms is the deflection due to bending, 
without taking shear into account ; the first term is the additional 
deflection due to shear. The ratio of the first to the second is 

6Eh 2 



oCL* ' 



This shows that for any usual ratio of h to L the additional 
deflection due to shear is only a small fraction of the whole 
deflection. 

The following case admits of still more simple treatment and 
is interesting as an example in which the shear strain may be of 
greater importance. Let the beam be of the I type, in which 
the shearing stress is practically all taken by the web and its 
intensity q over the section of the web is practically constant. 
With a single load W at the centre we have at all sections 



where A w is the (uniform) area of section of the web. This shearing 
stress produces everywhere a slope supplementary to the slope 
produced by the longitudinal strains in bending, of the amount 

., q TT 



L/ '1A W L/ 

Hence the supplementary deflection at the middle, where it is 
greatest, is 

,_ TT L 
U ~2A U .C r 

The deflection due to the longitudinal strains in bending is 
(by § 100) 

WD 
U ~±8EI' 

Taking the area of each flange to be uniform and equal to Ay we 
may treat I as practically equal to 

2 ' 

WL 3 

which makes u = ^ A ^ . 7 . 

24>EA f h 2 



DEFLECTION OF BEAMS : CONTINUOUS BEAMS. 143 

The ratio of the shearing to the bending deflection 

tf_6E Af h? 

u~ G ' A W 'L 2 ' 

As -jy is fully 15 in iron and steel, the shearing deflection may in 

this case form a considerable part of the whole, when the span is 
not a large multiple of the height and when the web is thin. 

In the considerations which follow, relating to continuous 
beams, the supplementary deflection due to shearing is not taken 
into account. 

106. Continuous Beams. A perfectly rigid beam resting on 
two rigid piers would be lifted off one or both of these if a third 
support were introduced. In a real beam the flexibility makes it 
possible for the load to be shared by more than two piers. A 
beam is said to be continuous when the number of its supports 
is greater than two. 

As a simple case we may first consider a beam of uniform 
section, uniformly loaded, resting on three equidistant piers at the 
same level. Imagine the middle pier to be removed, leaving an 
ordinary beam of span 2L. The deflection at the middle would 
then be (by § 100) 

as? EI -^ EI' 

In other words, this is the distance through which the middle pier 
would have to be lowered in order to relieve it of all share of the 
load. 

Now imagine the middle pier to be raised until it lifts the 
ends off their supports. The amount it must rise above the level 
would be equal to the deflection at the end of a uniformly loaded 
cantilever of length L, namely 

wL* 

*w 

The pressure on it would then be 2iuL. This pressure increases 
uniformly as the pier rises, from zero at a depth .,';, '■ below the 

level of the ends, to A 7 , r above the level of the ends. Hence 

8 LI 



144 



DEFLECTION OF BEAMS: CONTINUOUS BEAMS. 



when the middle pier is at the same level as the end piers the 
pressure on it is 

5 

. 2wL = %wL. 



5 

24 

5_-L.i 



24 



And the pressure on each of the end piers is consequently 

i(2-f)wZ = -fwZ. 

Another way of putting the matter is to regard the pressure F 
on each end pier as a single inverted load, acting on a cantilever of 
the length L to produce an upward deflection equal to the down- 
ward deflection which the load would produce on a cantilever of 

FL 3 

that length. The upward deflection due to F is r^.. Equating 

this to the downward deflection produced by the load, namely 

ttt^t i we have F = %wL. 
8EI 8 

If the middle pier were fixed at any assigned small height 
above or below the others it would evidently be easy to extend 
this treatment to find the proportion of load borne by it and by 
the other piers. The diagrams of shearing force and bending 




Fig. 91. 



DEFLECTION OF BEAMS : CONTINUOUS BEAMS. 145 

moment, when the piers are at the same level, are sketched in 
figs. 90 and 91. The diagram of bending moment is conveniently 
drawn by treating each half BG and BA as a cantilever fixed at 
J9, and loaded with an upward force F at the end along with a 
uniformly distributed load w. 

The points of inflection, where the bending moment is zero, are 
at a distance of \L from the middle pier. At any distance % from 
the middle pier the bending moment is 

M x = iwL(L- x )- w{L - x) \ 

The greatest negative bending moment occurs over the middle 
pier: its value is ^wL 2 . The greatest positive bending moment 
occurs at points § L from each end : its value is T fg wZ 2 . 

It may be instructive to treat this example of a continuous 
beam in a more general way, using a method which is suitable for 
application to other cases : — 

Let F be the (unknown) pressure on each of the end piers. 

Taking the middle point as origin we have 

M x = F(L -x) ^ — L 

T ( r, IVL\ /T1 rN WX 2 

= L[F- 2 -)-(F-wL)x--~. 
The slope 

[M 7 If/ ivL\ /rT T .x 2 wx' c 
l= JEi dx = El{ Lx [ F - 2J-( F ~ wL) 2- 6 

The constant of integration vanishes, since i = when x = 0. 

The deflection 

f. , 1 { T ( -n WL\X 2 ,_, T .X 3 WX 4 

u =j tda! = m{ L ( F - 2 jg-^-^e ~ 24 

The constant of integration again vanishes, since u = when 
^ = 0. 

Now if the piers are at the same level >/'=() when ./■=/,, 
and hence 



2 J ° v ' ±\ 

which makes F = \wL 

as before. 

E, s. M. L0 



146 DEFLECTION OF BEAMS : CONTINUOUS BEAMS. 

The student will find it a useful exercise to appl)~ this more 
general method to the case of a beam of 4 equal spans. He 
will find that each end pier bears a load of jqwL and each of the 
other two bears %^wL. 

107. Theorem of Three Moments. Whatever be the 
length of the spans and the mode of loading an equation can 
be found connecting the moments over any three neighbouring 
piers. This is Clapeyron's " Theorem of Three Moments," the 
algebraic expression of which does much to facilitate the solution 
in less simple cases than the one considered above. 

The theorem of three moments may be expressed in a gene- 
ralised form applicable to all modes of loading, but it will suffice 
for our purpose to consider the case of uniform loading only. 

Let A, B, C be any three consecutive piers (at the same level) 
in a continuous girder having any number of equal or unequal 
spans, uniformly loaded with a weight iu per foot run. The 
object of the theorem is to establish an equation between the 
three pier moments M A , M B and M c . The method we shall follow 
in getting the equation is to express the moment and the slope at 
B in two ways, by reckoning separately first from A and then 
from C. 

Taking the origin first at A we have for any point in the 
span AB 

M x = M A + F A m-^ (1). 

At B, where x = AB 

M B = M A + F A L AB -^ AB (2). 

Similarly by reckoning back towards B from C 

M B =M C +F C L CB -^D CB (3). 

Returning now to equation (1), since -j—, = -^ = =, we may 



write that equation thus 



EI d ^_ = M A + F A x^ (4). 



DEFLECTION OF BEAMS: CONTINUOUS BEAMS. 147 

Integrating to find the slope 

-,-, -,- ecu -. r -,-, x~ ivx .-, _ 

«r £ -*>+^ f — S-+0 < 5 >' 

where G is a constant of integration. 

Integrating again to find the deflection 

EIu = M A x ~ + F A ^- w ^ + Cx (6). 

The constant of integration is here zero, since u = when 
x = 0. 



Hence, since u = when x = L AB 

w 
24 



= \M A L\ B + \F A L\ B - ^ L\ B + GL AB 



on 

from which G = - \M A L AB -\F A L\ B + ^ D AB (7). 

Writing i B for the slope, or -=-, at B, we have by equation (5), 
since x is then L AB , 

EIi B = M A L AB + \F A D AB - iwL* An + G 
Substituting for G the value given in equation (7) we have 

Eli B = \M A L AB + \F A I? AB -\L\ B (8). 

In the same way, taking G as origin and reckoning back along 

GB we get 

w 
- EIi B = \M C L CB + \F C L\ B -^ L3 cb (9), 

the negative sign coming in on account of x being reckoned 
negatively. 

Equate these and eliminate the terms in F A and F c In 
substitution from equations (2) and (3), and we obtain an equation 
between the moments M A , M B and M c 

\M A L AB + M B L AB + bw&u = - \M Q L 0B - M a L a - \wL\ B ; 

or 

(M A + 2M B ) L An + (M c + 2M B ) L cn + ^ {L* AB + L\ fB ) - 0...(10) 

This is the theorem of three moments, expressed iii the 
comparatively simple form which applies to uniform Loading, It' 

10 2 



2M B L + 2M B L + -z- = 0. 



148 DEFLECTION OF BEAMS: CONTINUOUS BEAMS. 

there are n piers it yields n — 2 equations and the terminal 
conditions supply the two more which are required for solution. 
Usually the terminal conditions simply are, that the moments at 
the first and the last pier are zero. 

As an example of the use of the theorem we may first apply it 

to the case already treated, namely that of a continuous beam of 

two equal spans. Here M A = and M c — 0. Equation (10) 

becomes 

ivL' 3 

From which M B = — ~ — . 

wL 2 
Then, since M B = F A L — , F A = fwZ; as was found in § 106. 

* _ 

Similarly with three equal spans the moment at each inter- 
mediate pier is readily found to be , making the reactions 

j-qwL at each end pier and \^wL at each intermediate pier. 

Again, take the case of four equal spans. Here M A = M E == 0, 
M B = M D . 

Equation (10) gives 

*M B + M C 



and M B + 4if c + M D = 



2 ' 



2 



i-i u —wL 1 .. ,.. ,. — SivL 2 
from which M r = — — — and M B or M n = — ^^ — . 
c 14 B D 28 

The reactions at the piers are then found to be -J-J, §§, ||, f| 
and -^ of wL in each case*. 

108. Advantages of Continuous Beams. In a continuous 
beam the average value of the bending moment is much less than 
in a series of separate beams bridging the same spans and subject 
to the same load, and hence, by adapting the section to the 
moment at each point the continuous beam may be made much 

* For the graphic treatment of problems in continuous beams the student is 
referred to a paper by Professors Perry and Ayrton, Proc. Roy. Soc. 1879, and to 
Prof. Claxton Fidler's Treatise on Bridge Construction. See also Levy's Statique 
Graphique, Vol. ii. 



DEFLECTION OF BEAMS: CONTINUOUS BEAMS. 149 

lighter than the series of separate beams. But the advantage 
does not stop here : in the continuous beam the greatest values 
of the bending moments occur at and near the piers, whereas in 
the separate beam they occur at and near the middle of each span. 
Hence the heavier sections of the continuous beam are placed in 
positions where they are much less influential in causing bending 
moment. In long beams the weight of the beam itself becomes 
an important factor in producing bending moment: in a very long 
beam it is the chief factor. In such cases the advantage of 
continuity is specially great, on account of the concentration of 
weight near the piers and the comparatively light sections which 
are required towards the middle of each span. For short spans, 
where the externally applied load is the chief part of the whole 
load, the advantage of continuity is much less, especially when 
provision has to be made for moving loads. When moving loads 
pass over the beam the points of inflection change, and portions 
of the span are subjected to bending moments which change in 
sign as well as in amount. 

The advantage of continuous beams is practically much re- 
stricted by the possibility that the supports may yield and may 
thereby disturb the distribution of moments. A small amount of 
subsidence on the part of one of the piers may seriously alter the 
stresses and upset estimates of strength based on the assumption 
that the piers are on the same level. Even small errors in 
construction, whether in the level of the piers or the straightness 
of the built beam, are not without effect. 

When beams intended to be continuous have been in the first 
instance erected in separate spans they have of course to be 
connected in such a way as to secure effective continuity. Account 
must be taken of the flexure set up in each separate span by its 
own weight, and one or both of the distant ends must be lifted 
through a calculated distance before the near ends are joined over 
the pier, so that when the distant ends are let down again the 
moment due to the weight may be properly distributed. 

109. Combination of Cantilever "with Beams. In an\ 
continuous beam we might Imagine the beam to be cut at each 

point of inflection, and the parts to bo joined by a pin or other 

joint capable of resisting shearing force, but incapable of resisting 
bending moment. The stresses throughout bhe beam would be 



150 



DEFLECTION OF BEAMS : CONTINUOUS BEAMS. 



unaffected by this change. We should then have a system of 
cantilevers projected from the piers, united by beams between the 
ends of the cantilevers. 

Such a combination would however be free from the objection 
which has just been stated. Any subsidence of a support would 
not affect the distribution of stresses, because the points of 
inflection are now fixed. Moreover, they remain fixed when the 
loads change and the changes of bending moment and shearing 
force due to moving loads are readily calculated. The combination 
retains the main advantage and escapes the drawbacks of simple 
continuity. It has been used in some of the largest modern 
bridges, notably by Sir B. Baker in the great bridge over the 
Firth of Forth. 

110. Encastre Beam. An encastre or built-in beam is one 
whose ends are secured in such a way as to prevent any change of 
slope from taking place at the ends. The condition is not easy 
to realize in practice, and may be said to be never more than 
approximately realized, It will suffice for our present purpose to 
consider a beam whose ends are fixed horizontally so that they 
remain horizontal when the beam bends under load (fig. 92). 



Fig. 92. 




This may be regarded as equivalent to one span in a con- 
tinuous beam of an indefinite number of equal spans, similarly 



DEFLECTION OF BEAMS: CONTINUOUS BEAMS. 151 

loaded. Over each pier such a beam would lie horizontally. 
Hence we may apply the Theorem of Three Moments. 

Taking the case of a beam of uniform section uniformly 
loaded we have, in the expression of the Theorem given in § 107, 
M B = M c — M Ai and hence the expression becomes 

(3I A + 2M A ) L + (M A + 2M A ) L + 1 (2i») = 0, 

whence M A = — -=— . 

This is the bending moment which must exist at each of the fixed 
ends to keep the beam horizontal there. 

The forces which each support exerts upon the beam constitute 
(1) a couple whose moment is M A and (2) a vertical reaction equal 
to that part of the weight which comes on each end, namely \iuL. 
The first of these is the bending moment at the support, the 
second is the shearing force. 

At any point distant x from the end the bending moment 

x 
M x = M A + \wL .x—wx.^, 

the first term being due to the couple applied at the support, and 
the second term to the vertical reaction at the support. 

Hence M x = — ^- + -^-(L-x). 

To find the points of inflection we write M x = 0, which occurs 

when 

L 2 
x(L — so) = -75- 
o 

that is, when x = — ( 1 + -^ ) , 

which gives x= 0211Z and a? = 0"789Z/ as the distances of the two 
points of inflection from either support. 

At the middle, where the positive bending moment is a 
maximum, its value is 

— iuL 1 wL t r /A ?r/; 2 



,. — wJJ wL ( T L\ wis 



The diagram of bending moments is sketched in fig. 93. It 
may be described as the parabolic; diagram for a simple brain of 
span L, erected on a base ah which is the line of pier moments. 



152 DEFLECTION OF BEAMS: CONTINUOUS BEAMS. 

Instead of having recourse to the Theorem of Three Moments 
for finding M A we might have proceeded thus : — 



M x = M A + ™{L-x). 



The slope, 



1 
EI 



i = ^r^- I M x dx 



f / ivx \ 

= EI)\ Ma + T (Z ~ ^J dx 



M A x + 



wLx 



WX' 



El V A 4 6 

plus a constant of integration which must be zero since the slope 
is zero when x = 0. 

But i = when x = ~ , and hence 



^ + ^--^ = 0, 



i wZ 3 wZ s 
2 4 T6" " 48" 

'wL' 1 



from which il/. = 

A 12 

as before. 

It is interesting to compare the bending moment borne by the 
encastre beam with that borne by a beam of the ordinary kind. 
In a uniformly loaded beam of span L simply resting on end 

supports the greatest bending moment is — - . The encastre 

beam is consequently stronger, so far as the maximum moment 
caused by a uniform load is concerned, in the proportion of 12 to 
8, or 3 to 2. It should however be noticed that if any yielding at 
the supports occurs, which permits the beam to assume a slope at 
the ends, the advantage of the encastre form is quickly lost. 

To find the deflection of the encastre beam, uniformly loaded. 

we have 

/*., 1 f/—wL 2 x wLx 2 wx' 6 \ , 

« = r* - ei } { ~w- + ~t- - xj dx - 

At the middle, where the deflection is greatest, its amount is 

ImEi 

which is only one-fifth of the deflection in a similarly loaded 
simple beam. 



DEFLECTION OF BEAMS : CONTINUOUS BEAMS. 



153 



As another example, the case may be mentioned of an encastre 
beam of uniform section carrying a single load W at the middle. 
We may proceed as in the former case to find the moment at each 
support, or more simply infer it from this consideration : — In an 
indefinitely extended continuous beam of which the given encastre 
beam represents one span, the pier reactions are each equal to W. 
The system suffers no change by inversion : the bending moment 
over each pier is therefore equal to the bending moment under 
each load, and the convex portion of the beam over each pier must 
be of the same length as the concave portion under each load. 
Hence the points of inflection are at the distance \L from each 
support. The bending moment at the middle and at each support is 

WL 

— — and the diagram of bending moments has the form sketched 

in fig. 94. The deflection at the middle is readily found by 




adding that of a cantilever of length ^ loaded with \W at its end 
to that of a simple beam of length - loaded with ^^Y at its 



middle : its amount is 



WD 






192EI' 

The student will find it interesting to verify these results 
analytically, and to apply the same method of calculation to the 
case of a beam encastre at one end and resting at the other on a 
simple support at the same level. 



CHAPTER VIII. 



FRAMES. 



111. Frames. Among structures capable of bearing bending 
moments and acting as beams a highly important place is taken 
by frames. A frame is a structure composed of struts and ties. 
Although it may be subject to bending as a whole its separate 
parts or members are simply in tension or compression. This is 
because the members are attached to one another by joints which 
cannot transmit a bending moment and because the loads are 
applied at the joints. 

The simplest complete frame is a triangle (fig. 95). If such 
a frame rests on supports at A and B and carries a weight at G it 




Fig. 95. 

is serving as a beam although its three members are individually 
only subjected to tension and compression. We assume here that 
the members are connected by joints in which there is perfect 
freedom of angular movement — a condition which is approximated 
to when pin and eye joints are used. 



FRAMES. 



155 




Fig. 96. 




Fig. 97. 



112. Perfect, Imperfect, and Redundant Frames 

frame such as that sketched in fig. 96 
would be in equilibrium under a par- 
ticular distribution of loads, but not 
under any distribution. It is said to 
be imperfect, because the number of 
members is insufficient to make the 
frame preserve its shape when the 
loads vary. By adding one diagonal 
member (BC, fig. 97) it is converted 
into a perfect frame. The configura- 
tion now persists however the loads 
vary. Moreover, under any assigned 
system of loads the amount of pull 
and push on each member is deter- 
minate. 

Suppose, however, another member 
were introduced (AD, fig. 98). The 
amounts of pull and push in the 
several members are now indetermi- 
nate. The frame is now capable of 
being self-strained : that is, stresses 
may exist in the members apart from 

any application of loads. This was not possible in the perfect 
frame. A frame of this last kind may be described as having one 
or more redundant members. In practice it is sometimes useful to 
introduce redundant members for the following reason. Suppose 
in the frame of fig. 97 there were a great excess of load on joint A. 
The diagonal member BC would then be acting as a tie. But if 
the excess of load moved to joint B the diagonal would have to 
act as a strut. If BC were very flexible, and therefore incapable 
of acting as a strut, the frame would virtually be imperfect, but it 
could be made perfect by introducing the other diagonal AD which 
would then act as a tie. If both diagonals were present from the 
first, but both capable of acting as ties only, the frame would be 
well adapted for bearing an excess of load either at A or at />. In 
each case one of the two diagonals would simply go OUl of action 
and the other would serve to complete the frame. Members 
acting in this way — that, is, members capable of serving only as 

ties, or only as struts, and going out of action when a change in 




Fis. 98. 



156 



FRAMES. 



the distribution of the load tends to reverse the stress in them — 
are called semi-members. We shall have instances of their use 
later. In general, however, the frames which have to be con- 
sidered are those with simply the right number of members to 
be perfect in the sense explained above. 

113. Method of Sections. A bridge frame, such as the 
Warren girder of fig. 99, or the " N " girder or " Linville " truss 




Fig. 99. 

of fig. 100, may be regarded as a beam closely analogous to a solid 
beam of I section, but with this difference that the top and 



•d 



Fig. 100. 



bottom lines of members, corresponding to the flanges, are held 
apart by a network of bracing instead of by a continuous web. 

To find the stress in a top or bottom member we may 
calculate the bending moment M at a vertical section taken 
through the opposite joint. Take the section ab in Fig. 99. 
The stress in b prevents the right- and left-hand portions of 
the beam from turning about the joint a as a hinge. Conse- 
quently the amount of the stress in b is 

Mob 

h 



F h = 



where h is the depth of the beam at the section, measured from 
the joint to the middle line of the member b. In applying this 
method to a beam with vertical members, like that of fig. 100, it 



FRAMES. 157 

is convenient to think of the section as slightly inclined, so that it 
escapes coinciding with a vertical member. 

This method of sections is also applicable as a means of finding 
the stresses in what we may call the web members. Let F be the 
shearing force at any section such as cd, taken so as to cut an 
inclined member. The top and bottom members which are also 
cut by that section do nothing towards bearing the shearing force, 
for the stresses in them are wholly horizontal. Consequently 
the stress in the inclined member must have such a value that its 
vertical component is equal to the shearing force F at the section. 
Hence the stress in the inclined member is 

F 



cos 6 
where 6 is the angle the member makes with the vertical. 

In applying this principle to a frame with vertical members, 
the device of taking the section slightly inclined is again useful. 
Thus by taking cd slightly inclined as in fig. 100 we see at once 
that the stress in the vertical member it cuts is simply equal to 
the shearing stress reckoned by adding the loads on all the joints 
which lie to one or to the other side of the section so taken. To 
find the stresses in each inclined member of the " N " frame the 
section is taken vertical as in the Warren girder. 

The method of sections is specially convenient when the beam 
is of uniform depth. To find the stresses then involves little more 
labour than is required to tabulate the bending moments and 
shearing forces for successive panels of the frame. 

114. Graphic Process. Method of Reciprocal Figures. 

As an alternative to the Method of Sections the graphic method 
of Reciprocal Figures is in all cases practicable, and offers many 
advantages when the depth of the beam varies. It is the usual 
method of finding the stresses in the members of vool's, and is 
applicable to framework generally whatever be the directions of 
the applied forces. 

The method consists in drawing, superposed on one another, 
the polygons of forces for the several joints of the frame. It will 
be readily understood by reference to one or two examples. 

Take, for instance, the bowstring girder of fig. 101. For the 



158 



FRAMES. 



sake of generality we assume loads which are unequal and un- 
symmetrical. Find the reaction at each pier, either b} 7 taking 




moments about the opposite pier or by the graphic process of the 
funicular polygon described below in § 188. We adopt the method 
of lettering devised by Henrici and Bow, in which letters are 
placed in the spaces between members, and in the spaces separated 
by the lines of action of the applied loads, in a manner which the 
figure exemplifies. Thus we have the load BC, the pier reaction 
AB, the members EF, FG, and so on. Similarly the joints are 
named by the letters round them, thus the girder rests on the 
left-hand pier at the joint EAF. 

Begin by drawing the polygon of forces for that joint (fig. 102). 
Taking any convenient scale of forces, set out the known force 




Fig. 102. 



FRAMES. 159 

EA (fig. 102) and find the forces AF and FE by drawing lines 
parallel to these lines in the frame. The triangle EAF in fig. 102 
is the polygon of forces for the joint EAF in the frame. The 
forces in the triangle have the directions EA, AF, FE. This 
shows that the member AF in the frame is a strut, and the 
member FE is a tie. Mark them so by arrows, and go on to 
draw the polygon of forces for another joint in the frame. A joint 
must be taken at which there are not more than two unknown 
forces: hence the next to be taken is the joint FAG. We use 
the line FA already drawn in fig. 102, and complete a triangle on 
it by lines parallel to the members AG and GF. These give the 
forces in those members. The next joint is DEFGH. The forces 
DE, EF, FG are already known, GH and HD are to be found. 
Take a point D on the vertical line through E in fig. 102, at a 
height above E which represents on the scale of forces the load 
DE : then we have the polygon DE, EF, FG, completed by lines 
GH, HD parallel to the corresponding members of the frame. By 
proceeding in the same way from joint to joint the complete 
diagram of fig. 102 is built up. It is a group of superposed 
diagrams of forces for the several joints, each line serving twice 
over, for the stress in each member acts as a force at each of the 
two joints which the member connects. 

The lines in the two figures, the frame and the force diagram, 
are severally parallel, and each group of lines which meet at a 
point in the one form a closed polygon in the other. For this 
reason the figures are described as "reciprocal." 

When drawing the force diagram it is important, at each joint. 
to follow the same order in taking the forces. In the example 
given above the forces are taken "clockwise" round the joint. 
At each joint all the known forces are dealt with first, in drawing 
the polygon, and the polygon is closed by lines parallel to the 
unknown forces, the number of which must therefore not exceed 
two. 

115. Examples of the Method of Reciprocal Figures 
This graphic method of determining bhe Btressea in the members 
is applicable to frames of all kinds, Loaded in any manner. The 
externally applied forces Deed Dot be vertical. In the diagram of 
forces they form a closed polygon, since bhe frame as a whole is 
in equilibrium under them. When bhe loads and reactions of bhe 



160 



FRAMES. 



supports are vertical, the sides of this polygon coalesce into a 
vertical line : the line of reaction (directed upwards) then coincides 




Fig. 103. 



.# 




FRAMES. 



161 



with the line of loads (directed downwards). In the diagram, 
fig. 102, the line of loads is BE; the reactions are EA and AB. 

Figs. 103 and 104 exemplify the method as applied to a crane 
or bracket supported by a socket at the foot and a horizontal tie 
CB. The thrust on the socket need not be determined before- 
hand : it is found when the force diagram is drawn. In drawing 
the diagram we begin with the joint GAD, then take the joint 
DAE, and so on. The triangle ABC is the polygon of the external 
forces, and AB determines the direction and magnitude of the 
thrust at the socket. 




Fig. 105. 




B. S. M. 



11 



162 



FEAMES. 



Fig. 105 is a simple example of a symmetrically loaded roof. 
In such a case it suffices to draw half the diagram of forces, but 
by completing it, as in fig. 106, we have a useful check on the 
accuracy of the work. 

116. Use of the Funicular Polygon in finding the 
Reactions at the Supports. In fig. 107 the roof frame is 




Fig. 107. 

unsymmetrical, and this example will serve to show how the 
reactions at the supports may be graphically determined by the 
use of the funicular polygon. The funicular polygon is an 
imaginary chain which would be in equilibrium under the given 
loads. When the ends of such a chain are held apart by an 
imaginary bar they produce at the supports which carry them 
the same reactions as the loaded frame produces. Hence by 
finding the reactions due to the imaginary loaded chain we find 
those due to the frame itself. 

To draw the funicular polygon, draw the line of loads BCDEFG 
(fig. 108). Take any pole and join it with the points B, C, D, etc. 
Then from any point in the line of the reaction AB draw a line OC 
(fig. 107), parallel to the line OC of the force diagram, to meet 
the line CD along which the load CD acts. Complete the funicular 
polygon by lines OD, OE, OF parallel to the corresponding lines in 



FRAMES. 



163 



the force diagram, and draw the bar OA joining its ends. Then 
in the force diagram (fig. 108) draw OA parallel to OA in fig. 107. 




It divides the line of loads in a point A such that A B and GA 
are the reactions which were to be found. 

Having determined the reactions, the force diagram for the 
frame is readily drawn. It is shown on the left-hand side of the 
line of loads in fig. 108. 

In farther illustration of the method we may take a case 
where the loads are not all vertical. Suppose wind to act on a 




Fig. 101). 



11—2 



164 



FRAMES. 



frame making the loads CB, BE, EF, and FG inclined as sketched 
(fig. 109). In that case one or both of the pier reactions must 
have a horizontal component. Let us assume that the roof is 
anchored at the left side, but that the reaction GA is vertical. 
Having drawn the line of loads BCDEFG (fig. 110) and selected 
a pole 0, proceed to draw the funicular polygon, starting from 
the left-hand side, where the direction of the reaction is not 
known. This determines the lines OG, OB, OE, OF, OG in 
fig. 109. The joining bar OA is added, and the line OA is drawn 
parallel to it in fig. 97, to meet a vertical reaction line from G. 




This determines GA, which is the reaction at the right-hand end. 
Then the reaction at the left-hand end is determined, in magni- 
tude and direction by the line AB (fig. 110), which completes the 
polygon BGGA of the external forces. The student will complete 
for himself the force diagram of the frame. 

117. Special Cases. In some frames a difficulty presents 
itself in the drawing of the reciprocal figure or force diagram, 
which has to be overcome by a suitable artifice. Consider, for 
example, the frame of fig. 111. Beginning with the joint ABCL 
the reciprocal figure is readily drawn for that joint, and for LGBM, 



FRAMES. 



165 



and for ALMN. But we are then confronted by the difficulty 
that there are more than two unknown forces at either of the 




neighbouring joints AN OR or NMDEPO. In order to proceed 
with the reciprocal figure, we determine, independently, the 
stress in AR. This may be conveniently done by the method of 
sections, taking moments about the top joint, or it may be done 
graphically by the following device. The stress in AR depends 
only on the loads and on the skeleton outline of the frame 
(fig. 112): in other words, it is independent of the character of 




the bracing within the panels Y and Z. Hence we may omit 
this bracing altogether and refer the loads to the top and bottom 
joints of each rafter as in fig. 99, and then draw as much as ifl 
necessary of the reciprocal figure for that simple frame to find 
the stress in AR. This is done in fig. LIS. Another method ifi 
to substitute for the actual bracing in the panels Fand £a form 



166 



FRAMES. 



of bracing which escapes the difficulty, as in fig. 114 : then by 
drawing part of the reciprocal figure for this altered frame we 




Fig. U3. 



-7 



find the stress in AR, which is not affected by the change. 
Having found the stress in AR there is no difficulty in completing 




the reciprocal figure for the original frame. It is sketched 
complete in fig. 115. 

118. Use of Semi-members in Bridge Frames. Counter- 
bracing. In a frame such as that sketched in fig. 116, the 



FRAMES. 



167 



diagonal members in the panels act as ties if the bridge is 
symmetrically loaded. To the right of the centre the shearing 




Fig. 115. 



stress for any system of symmetrical loads is positive : hence at 
any section on the right of the centre the diagonal cut by that 




Fig. 116. 

section is pulling, to hold down the part of the frame lying to the 
right of that section. But if the loading is unsymmetrical the 
shearing force in any panel may change its sign, and in thai 
case the diagonal would have to act as a strut. To avoid this 
necessity the other diagonal would in general be introduced 
wherever such a reversal is liable to occur. The two diagonals 
then form semi-members (§ 112), one being in action only when 
the loads are such as to make the shearing force positive, and the 
other only when the shearing force is negative. Panels thus 
treated are said to be " counterbraced." 



168 



FRAMES. 



In practical cases the load on a bridge consists of two parts ; 
one is the steady load due to the weight of the structure and of 
the roadway, the other is the variable or rolling load which passes 
over the bridge. If all the load were of the latter kind the 
shearing stress would be liable to reversal in every panel during 
the coming on and passing off of the rolling load, and in that case 
every panel would have to be counterbraced. But the presence 
of steady load tends to prevent this reversal from happening, 
except in the central panels. Consequently the number of panels 
which require counterbracing depends on the proportion of the 
rolling load to the steady load. 

In a beam loaded with a steady load w per foot run the 
diagram of shearing force is that sketched in fig. 117. If a 
rolling load of w' per foot run be supposed to come on from one 
end, it causes the shearing stress to take the maximum positive 
and negative values shown in fig. 118. Under the combined 
action of both loads the sign of the shearing stress suffers change 




Fig. 118. 



throughout the portion ab of the beam, a and b being taken so 
that ae — ac and bd = bf. Outside of the limits ah there is no 
change in the sign of the shearing stress as the rolling load passes 
over the beam. 

Hence in a frame beam similarly loaded, enough panels would 
have to be counterbraced to include the region ah With a frame 
in which the load is assumed to act at the joints there would be 
stepped lines in the diagram of shearing force, instead of -the 
continuous lines shown in figs. 117 and 118. The steps follow 



FRAMES. 



169 



the same general outline, and in applying this construction to 
find the number of panels which should be counterbraced no 
inconvenience is caused by treating the load as continuous. 

Another common instance of the use of semi-members is found 
in frame piers. The panels of the pier are counterbraced so that 
one or the other diagonal will be in action when a horizontal load 
comes from one side or the other, such as would arise from the 
action of wind on the pier and on the structure which the pier 
carries. 

119. Superposed Frames. Two or more frames may be 
superposed to form a compound frame, the members of which 
fulfil distinct functions in each of the component frames. Thus a 
double Warren or lattice girder (fig. 119) is obtained by superposing 
the frames of figs. 120 and 121. Each of these is readily examined 




Fig. 119. 




Fig. 120. 




Fig. 121. 

by the method of sections or the method of reciprocal figures, and 
the members which arc common fco both frames have to bear 
stresses equal to the sum of (hose determined for the component 
frames separately. The Kink truss (fig. L22) and the Bollman 



170 



FRAMES. 



truss (fig. 123) are other examples of compound frames, the only 
common members in them being those which make up the top 
boom. 




Fig. 122. 



f f I I Y { I 




Fig. 123. 

120. Effects of Stiff Joints. In the ideal frame the joints 
are perfectly flexible, in the real frame they are frequently stiff. 
Where pin and eye joints are used the condition of perfect 
flexibility is approached — though on account of friction at the 
pins it is not quite realized. But in many frames no attempt is 
made to give the members freedom of relative turning at the joints. 
Rivetted bridge-work and ordinary timber roofs are familiar 
instances of frames with stiff joints. When the joints are stiff 
the stresses are, strictly, indeterminate, for the frame may then 
be self-strained although it has no redundant members. Further, 
the stresses which are caused by external loads do not then admit 
of exact determination : the stresses cease to be necessarily axial, 
and the tension members are liable to be bent as well as pulled. 
The maximum intensity of stress in the tension bars may therefore 
be expected to exceed the value which would be found if the joints 
were flexible. On the other hand the compressed members are, 
for reasons which will appear in the next chapter, strengthened as 
well as stiffened by fixing their ends. On the whole the frame 
with stiff joints will generally be stiffer for small loads than the 
other : that is, its elastic yielding will at first be less. It may. 
however, be expected to reach its elastic limit sooner, although if 
loaded to rupture it may stand a greater load. 



CHAPTER IX. 



STRUTS AND COLUMNS. 



121. Instability under Compression. A piece under 
compression differs from a piece under tension in this important 
respect, that if the distribution of the stress is for any reason not 
strictly uniform, the yielding of the piece tends to increase the 
inequality instead of reducing it. When a column is compressed, 
if there is at any stage in the process an inequality in the intensity 
of stress on two sides, the side that is more strongly stressed yields 
more than the other, and the column bends in such a manner as 
to bring the resultant thrust nearer to the more stressed side, with 
the result that the inequality is made greater than before. 

Some initial inequality in the distribution of the stress must 
in all cases be expected. It may arise from a want of perfect 
straightness to begin with, or from unsymmetrically shaped ends, 
or from other causes which make the loading not perfectly axial. 
Or it may arise from non-uniformity in the elasticity of the column 
itself, due to a want of homogeneity in the material, or from the 
casual application of some force distinct from the load. 

The influence of this bending is more felt in long columns 
than in short ones, but it may be traced in the crushing even of 
a block whose length is four or five times its diameter. Let 
such a block be tested in compression and it will bo found to 
yield with a smaller total load than would bo anticipated from 
the known crushing strength of the material, and in yielding it 
will be seen to bend. Experiments intended to determine crushing 
strength are consequently made in general on blocks which are only 
about one and a half or two diameters long, 



172 STRUTS AND COLUMNS. 

In practical cases a column or strut is usually so long, in 
comparison with its transverse dimensions, that the tendency to 
bend under a longitudinal thrust is the main consideration 
affecting its strength. We accordingly consider first the case of 
a very long column, the theory relating to very long columns 
being afterwards modified to make the results applicable to 
columns of ordinary length. 

122. Bending of Long Columns. Euler's Theory. 

Consider a strut or column whose length is very great in 
comparison with its transverse dimensions. Assume it to . 

be originally straight and of uniform section, to be loaded 
axially, and to be symmetrical as to elasticity. We shall 
further assume it to have round ends, in other words, that 
it is free to bend along its whole length, as in fig. 124. 

Suppose that while the column carries an end-load it 
is caused to become slightly deflected (say by the appli- 
cation of a side force which is immediately removed). 
In consequence of the deflected position of the strut 
there is now a bending moment acting at every section. 
If the end-load has a certain value P the deflection will 
persist : if it has a smaller value the strut will straighten 
itself; if it has a greater value than P the deflection will 
increase. We have to find the critical value P which 
will just serve to keep the strut from straightening itself. 

Taking the middle point of the chord as origin, the 
bending moment at any section distant y from is Pu, 
u being the deflection there. Since the strut is in 

.,., . , , 1 d 2 u , 

equilibrium the curvature, -~ or -7-^, must be propor- 
tional at every section to the bending moment, and 

dru Pu 

where I is the moment of inertia of the section about a central 
axis perpendicular to the plane in which curvature has taken place. 
The negative sign in this equation arises from the fact that the 
centre of curvature lies on the negative side when the deflection 
is positive. 



124. 



STRUTS AND COLUMNS. 173 

Assuming the section of the strut to be uniform., the solution 
of this equation is 

u = til cos y a/ -prf, 



where v^ is the deflection at 



* 



Now u = when y = - ' L being the length of the strut, and 



hence 



L /T 

2 



^v wr°> 



from which 2\ EI = I' 

Hence the value of P is 

This is the amount of end-load which is just sufficient to hold 
the strut bent once curvature has been produced. It is important 
to notice that P is independent of u x : in other words, the same 
force will serve to keep the strut bent whether the curvature is 
small or not so small. This is equivalent to saying that the strut 
is in neutral equilibrium under the critical load P. Under any 
smaller load it would be in stable equilibrium ; under any greater 
load its equilibrium would be neutral, for the curvature once 
induced would increase without limit under the action of the load. 

Hence this value of P must be taken as the limiting load 

ir 2 EI 

which the strut can support. In this sense the expression -=j 

measures the strength of the strut. This is Euler's Theory of the 
yielding of struts. It is valid only when the strut is long and 
when the loading is perfectly axial, and the strut is perfectly 
straight and perfectly symmetrical in respect of elastic quality. 

In such a case, the strut on being loaded would show no 

* The general solution of the equation is 

U=A cosy *y -j + BsinijA^/ = . 

where A and B are constants whose values are to be determined from tin 
conditions of the particular case. In the case of fig. 124 the QOnditiona arc thai 

ii = -u 1 when y = 0, and that uaOwhen j/== =r, and also when // = 

Hence I>=0 and A = ?/, . 



174 STRUTS AND COLUMNS. 

permanent bending (after any casual side force had acted) until 
the critical load P was reached. Under that load it would 
maintain any bending that might be given to it. With the least 
further increase of load it would give way completely. 

L /P~ 
The condition cos -=r a / -=> = 



is also satisfied when 



2'V EI* 

L /~P r __mr 

2 V EI~ 2 ' 

n being any integer. This gives a series of higher values of P, 

namely 

n 2 ir 2 EI 

L 2 ' 

The physical interpretation of these val ues is that they are the 

critical loads for a strut bent in segments, the lengths of which are 

L L 

— , — , and so on. 

L o 

These modes of bending do not however need to be considered 
in dealing with the strength of struts. 

123. Fixed and Free Ends. The above theory requires 
modification when the ends of the strut are held fixed so that 
they are forced to remain parallel to the direction of 
the thrust when the strut bends. This state of things 
is illustrated in fig. 125. The line of thrust PP then 
passes through the points of inflection B, D : by deviating 
from the ends A, E, it supplies the bending moment 
required to maintain finite curvature there. The 
section of the strut being, by assumption, uniform, 
the condition of equilibrium is identical at points 
between B and A and at corresponding points between 
B and C. A corresponding symmetry holds for points 
above and below D, Hence the points of inflection 
are at one-fourth of the length from each end. The 
yielding of the strut takes place under that critical 
load which would cause yielding in a round-ended strut 
of the half-length BD — namely when 

7T 2 EI 4,7T 2 EI 



P = 



BD 2 " L 



In this case the ends are fixed in such a manner 
that their position as well as direction is maintained. 



STRUTS AND COLUMNS. 175 

If, however, the conditions were such that there was freedom 
on the part of the top of the column to move sideways, the 
direction of the ends only being constrained, then the strut as a 
whole would be represented by the portion AC of fig. 125, and in 
that case 



IT 



EI 



p = » • 

where L stands, as before, for the actual length of the strut. 

When one end is fixed and one is free to turn, but not free to 
move sideways, the bent strut takes a form approximating to the 
curve BGE of fig. 125, and the critical load is that which would 
correspond nearly to a round-ended strut of § the length, or 

9tt 2 EI 
4Z 2 ' 

Finally, when one end is fixed, and the other is free not only 

to turn but to move sideways, the condition of the whole strut is 

represented by the curve BE of fig. 125, and the critical load is 

that which would be borne by a round-ended strut whose length 

is 2Z ; in other words 

p = ^EI 

4Z 2 ' 

Euler's formula may be expressed to suit all the cases in 
this form 

117T 2 EI 

I 2 ' 

where n is a constant depending on the manner of attachment of 
the ends. 

The cases are summarised below : 



1. Both ends round. 

2. Both ends fixed in direction and 

position. 

3. Both ends fixed in direction. One 

end only fixed in position. 

4. Both ends fixed in position. One 

only fixed in direction. 

5. One end fixed in direction. The 

other end round and free to 
move sideways. 



Critical load P by 
Euler's theory. 


It 


tt'EI 


T 


L l 


1 


W'EI 


4 


7T*E1 

V 


1 


Ott-v;/ 


9 


4/;-' 


4 


TT-'AV 


1 


\l. 


4 



176 STRUTS AXD COLUMNS. 

It may be noted in passing that a strut may be attached in 
such a way that it has round euds with respect to one direction of 
bending, and fixed ends with respect to the other. 

Euler's theory requires important modification when applied 
in practical cases. It serves however to show the primary im- 
portance of giving a strut a form of section in which the moment 
of inertia is large. 

124. Modification of Euler's theory to meet practical 
conditions. We have next to consider what modifications are 
imposed on Euler's theory by the conditions which hold in the 
case of real columns. 

In the first place it is clear that a very short column will not 
yield in the manner contemplated in Euler's theory, but will yield 
by direct crushing. Its strength will depend, at least mainl} 7 , 
on the crushing strength of the material — a term which does 
not enter into the Euler formula. For a very short strut the 
application of that formula would give a load greater than the 
simple crushing strength of the strut. In the ideal very short 
strut, where bending plays no part in causing failure, the breaking 
load would be 

fcS, 

where / c is the crushing strength of the material and S is the area 
of section. In the ideal very long strut of Euler's theory the 



breaking load is 



7T- 



EI 



But the breaking load must alter in a continuous manner as 
we pass from very short to very long lengths, and an equation 
which will express a continuous relation between the two is to 
be found. 

If we write 

feS_ 



P = 



1 +fcS ^m 

we have a relation between P and L which is continuous and 
which makes P=f c S when L is indefinitely small, and also 

2 T^T 

makes P = — ^— when L is indefinitely great. In other words, 



STRUTS AND COLUMNS. 



177 



it makes P approximate to f c S when the strut is very short, and 



to 



IT 



EI 



when the strut is very long. 



Further it gives, for struts of intermediate length, a value of P 



which is considerably less than the value 



TT 



EI 



D 



corresponding to 



Euler's ideal strut. This is as it should be, for the ideal conditions 
of perfect straightness, perfect symmetry of elasticity, and perfect 
centrality in the application of the load are never realised in 
practice, and any deviation from these conditions makes the actual 
breaking load less than the ideal load of Euler's formula. 

Fig. 126 illustrates the curve which represents the relation of 
strength to length as expressed by this formula. The height of 



A 


\B 








Q 










< 

o 




Fig. 


126. 




-J 


D\ \ 








o 










z 










X 










< 










Hi 










c- 










OQ 








===—2 



LENGTH 

A is f c S. The line BC represents values of P found from Euler's 

if' EI 

equation P= Jt> . The continuous curve ADE, which rises 

nearly but not quite as high as BC when the length is very 
great, is got from the equation stated above. 

If now we take the results of any group of experiments made 
to determine the breaking load of struts of various lengths, of the 
same material and the same cross-section, wo find that the points 
representing such experiments lie, in general, fairly well on a 
continuous curve resembling ADE, The experimental curve will 

however lie lower than the curve BO even when thfl struts are 

long, because of the weakening which is Introduced by deviation 
from the ideal conditions assumed in Euler's theory. 

E. S. M. 1 2 



178 STRUTS AND COLUMNS. 

Moreover, these sources of weakness are irregular, and cannot 
be expected to show themselves equally in all the experiments. 
One strut will be more homogeneous than another, straighter, or 
more strictly axial in its loading. Hence the results of experiments 
are found to lie irregularly about such a curve, and it is only at 
the best a roughly approximate expression of them that can be 
given by any formula. 

The formula may be made to agree most closely with experi- 
mental results by treating it as empirical, and adjusting the 
constants. In other words, the constants in the numerator and 
denominator are to have such values assigned to them as will 
make the values of P agree most closely with experiments on 
struts of the same material but of different lengths. 

Thus we may write it in the form 

P fS 

1 + 4c ~y 

and then select values of f and 4c to suit the available ex- 
periments. 

This may also be written 

/ 



p 



1 + 4c ¥ 



where p is the breaking load expressed per square inch of the 
section, and k is the radius of gyration of the section with respect 
to the axis about which bending is most likely to take place, 
namely the axis about which / is least. 

This is for struts with round ends. When the ends are fixed 
we have 

/ 

1 + c ¥ 

and generally we have for the other cases distinguished in § 123, 

4c L 2 

— as the coefficient of -j- , where n has the values stated there. 

to tC 

A formula of this kind was first put forward by Prof. Lewis 
Gordon, who based it on a suggestion of Tredgold, and it was 



STRUTS AND COLUMNS. 179 

adopted with some modification by Rankine. In the form which 
has just been given it is generally known as Rankine's formula. 

In Gordon's original formula, the ratio of length to least 
breadth of section was used, instead of the ratio of L to k. This 
alters the constant in the denominator ; thus for a strut with fixed 
ends, 

/ 

pss : "/Xv' 

l + a 



b. 

where b is the least breadth of section and a is a constant 
depending both on the material and on the type of section. 

Rankine's modification of Gordon's formula makes it applicable 
to columns of any section. 

The constants a and c are connected by the equation 

c _^ 
a~b 2 ' 

b 2 
In a strut whose section is a solid rectangle k 2 = r-= and 

consequently a = 12c. Other cases are tabulated below. 

k 2 



Form of section 



b 



Solid rectangle 3-9 • 

Thin hollow square ^ . 



Solid circle 

Thin hollow circle 

L, T, or cruciform section with 



J_ 

16* 

1 

8* 



18' 



equal sides, each equal to b, ~r . 

H section, with equal web and 

flanges 

Among the most authoritative experiments from which the 
values of the constants maybe deduced are those oi' Hodgkinson*. 
The following table gives some of his results for solid rectangular 
columns of wrought-iron having flat and well-bedded ends. The 

* Phil Trans. Hoy. Soc. IS 10. 1857. 

I _ w 



180 



STRUTS AND COLUMNS. 



behaviour of these columns may be taken as that of columns with 
fixed ends. In all the instances cited here the least width was 
exactly or approximately 1 inch (generally 1'023 inch), and 
the other transverse dimension was approximately either 1 or 3 
inches. The exact dimensions of the section are used in calcu- 
lating the breaking load per square inch and the ratio of length L 
to least breadth b. 



Approximate 

dimensions 

of section 


Length 
in inches 


Eatio 
L 

1 


Breaking load 

in tons per 
square inch, p 


lxl 


7-5 


7-3 


21-7 


1" x 1" 


15 


14-6 


154 


1" x 1" 


30 


29-3 


11-3 


-t II nil 

1 x 6 


30 


30 


13-2 


r x i" 


60 


58-6 


7-7 


1" x 3" 


60 


60 


8-1 


-in -in 

lxl 


90 


88 


4-35 


1 " 9" 

1x6 


90 


90 


4-42 


17) O'/ 

lxo 


120 


120 


1-91 



Fig. 127 shows these values of the breaking strength, in 



2D 








A 

\ 












X 

o 

- 20 
LU 

cc 
< 

D 
Gf 

CO 









\ 




• ^ 








c 


\ 


1 














cc 
uj 15 




sP\ 
















o. 






\ 














co 

z 
O 

h 

Q 10 

< 

O 

_l 




























\^ C 


) \. 








o 

z 

cc 

03 
































^*^>^ 






















""""-* 


B 
D 

) 



20 40 60 80 

RATIO OF LENGTH TO BREADTH. 

Fig. 127. 



100 



120 



STRUTS AND COLUMNS. 181 

relation to the ratio of length to least breadth ( t ) • The separate 

observations are shown by small circles, and the dotted curve is 
sketched to lie as well as possible among them. The curve CD, 
shown by a full line, represents the breaking strength given by 
Gordon's formula, 

P= — ^7xv> 

where the constants f and a have the values assigned to them which 
Gordon found to represent best the results of these and other 
experiments by Hodgkinson on rectangular wrought-iron columns, 
namely 

/= 16 tons per sq. inch, a = goVo* 

The corresponding value of c, the constant in Rankine's 
formula, would be -g^joo". These constants are usually accepted 
for wrought-iron. 

Fig. 127 will serve to illustrate the necessarily rough character 
of experiments or calculations on the strength of columns. The 
results of experiments differ somewhat widely amongst themselves, 
and the Gordon or Rankine formula does not express even the 
average of the experimental results with any great precision. The 
agreement between the results of experiment and the curve CD 
of the figure is good for struts whose lengths range from say 20 
to 100 times their breadth, but for shorter struts there is con- 
siderable discrepancy, of such a kind that the formula errs on 
the side of safety. 

It is interesting to compare these experimental values of the 
breaking strength with those which would be found, in an ideal 
strut, on Euler's theory. We may take 12000 tons per square 
inch as a probable (rather low) value of E in wrought-iron. 
Then for an ideal column 1 inch square, with fixed ends, Euler'a 
theory would give 

_ 4tt-EI _ 39500 
P ~ I? ~ L* 

in tons per square inch, L being in inches. The curve AB in 
fig. 127 gives the values of p derived from this. It shows, when 
compared with the results of the experiments, How wide of the 



182 STRUTS AND COLUMNS. 

mark Euler's theory would be if applied to struts of moderate 
length. 

What happens when the real strut is tested under compression 
is that owing to the departure from the ideal conditions of 
symmetry in form and quality and load it bends, at first slightly, 
and each addition of load is associated with a finite increase in the 
deflection. 

In this bent state the distribution of stress is not uniform 
because the resultant passes away from the centre, and as bending 
proceeds the stress on the off-side may change to tension. Failure 
ensues when the greatest compressive stress on one side or the 
greatest tensile stress on the other exceeds the limit of elasticity. 

125. Values of the Constants in other materials. The 

values of the constants derived from experiments on cast-iron 
columns with flat ends are 

/= 36 tons per sq. inch, c = g^Vo- 

Thus for cast-iron pillar, with flat ends, when the section is a 

hollow circle, we have 

36 



^800 \d. 



where d is the external diameter. 
When the section is a solid circle 

36 
P = ^W 



1 + 400 \dj 

Experiments on steel columns have been carried out by 
Mr Christie*, who gives a table of average results. Professor 
Fidlerf finds that these are expressed with a fair degree of 

accuracy by Rankine's formula, for ratios of -j- greater than 20 

and less than 200, taking the following values for the constants : 

For mild steel, /= 21*4, c = ^oho- 

For hard steel, /=31'2, c= 2 oooo * 

* Tram. American Inst, of Civil Engineers. 
t Treatise on Bridge Construction. 



STRUTS AND COLUMNS. 183 

It is important to notice that the formulas of Gordon or 
Rankine with these various empirical constsfnts are not to be 
accepted as applicable beyond the limits of length reached in the 
experiments on which the constants are based. If we were to 
apply them to longer struts they would give a greater strength 
than is compatible with Euler's theory, whereas a real strut of 
great length is certainly weaker than Euler's ideal strut. 
Rankine's formula, for a strut with fixed ends, 

/ 

P-- -j}, 

gives values of p which approximate more and more closely to 

/ k> 
c' L 2 

the greater the ratio of L to k becomes. 

Now Euler's theory gives for the breaking strength of the long- 
ideal strut 

P WEI . 9r7 k* 

S = l3ir = * 7rE -Lf 

It is certain that the strength of the actual strut is somewhat less, 

and hence 

f 

- should be less than WE 

c 

if the practical formula is to be applicable to very long struts. 
In fact, however, the constants which are usually accepted do not 
satisfy this test. Thus for wrought-iron or steel 4nr' 2 E ranges 
from about 480000 to 520000, but in wrought-iron 

* is 16x36000=576000 
c 

if we take the received values of these constants. For stool the 

f 

constants stated above make - about 620000 or 640000. These 

c 

considerations only strengthen what has been already said, that 

a formula expressing fche results of experiments od stmts is 

essentially empirical, and has Little or DO value when extended 

beyond the limits of experiment. 



184 STRUTS AND COLUMNS. 

126. Struts with Lateral Load*. When, in addition to 
thrust at its ends, a strut carries lateral load which produces 
bending moment, the strut becomes laterally deflected and the 
whole bending moment at any section is made up of the bending 
moment due to the lateral load, together with the bending 
moment due to the end thrust. Let fi be the bending moment 
due to lateral load at any section, and u be the deflection there. 
Then the whole bending moment in that section is 

M = fi + Pa, 

and therefore the curvature 

d^ = -EI i/M + Pu) - 

To integrate this we must be able to express /iasa function of x, 
which is the distance from the origin in the direction of the strut's 
length. 

As a particular case, suppose that the lateral load is distributed 
more or less uniformly. The diagram of bending moment, due to 
a uniformly distributed load alone, is a parabola which is not far 
from coincidence with a curve of sines. Thus if we express the 
moment due to the lateral load in the form 

f 1 = i WL cos -y- 

where W is the whole of the distributed load, we get values 
which are nowhere widely different from the values which would 
be given by uniform loading. At the middle, where x = 0, this 
expression makes fi = J WL, and at the ends fi = 0. These are 
correct for uniform loading.. At points between the middle and 
the ends it gives values which are slightly less than those which a 
uniform loading would produce. 

Expressing /ul in this manner we have 

d 2 u Pu WL irx 

d^ + EI + 8EI C0S 77=°> 



which gives u = 



A WL cos -=- 
° L 

EI$-P 



as the equation for the deflection of the strut. 

* See a paper by Prof. Perry, Phil. Mag., March, 1892. 



STRUTS AND COLUMNS. 185 

At the middle, where this deflection is greatest, its value is 

WD 

Ui ~8(tt 2 EI-PD)' 

The greatest bending moment is 

WL PWD 

fl t + Pll, = —q- + 



8 *{tt*EI-PD) 



W L / , P 

1 + 



8 1 ^- p 

7r' 2 EI . 

The quantity is the load which would cause instability, 

by Euler's theory. Calling it Q we may express the greatest 
bending moment of the laterally loaded strut in the following form : 

1Ul 8 \Q-P)- 

Having found the greatest bending moment we may readily 

proceed to find the greatest intensity of stress. It is made up of 

the stress which the load P alone would cause, if acting axially, 

together with the stress which is produced by the bending 

moment. Taking the middle section, where the stress is greatest, 

let y x and y 2 be the distances from a central axis through the 

centre of gravity of the section to the inner and outer edge 

respectively. Then the stress due to bending alone produces a 

M y 
compression equal to — j l at the inner edge, and a tension equal 

M y 

to --p- 2 at the outer edge. Hence the greatest intensity of com- 
pressive stress, namely at the inner edge of the middle section, is 

M xVl P 

where A is the area of the section, and / is its moment of inertia 
about the axis with respect to which bending occurs. The 
greatest intensity of tensile stress occurs at the outer edge of the 
middle section, and its value is 

My* __ P 

I A' 

From these expressions (hat value of / } may be calculated 
which will cause the greatest stress t<> reach an assigned limit 



186 STRUTS AND COLUMNS. 

when the lateral loading is known ; or alternatively the amount of 
lateral loading can be determined which, in conjunction with a 
given thrust P, will cause the greatest stress to reach an assigned 
limit. The theory also serves, of course, to test the suitability of 
assumed dimensions of section when the end thrust and lateral 
load are both assigned. 

A practical case occurs in the coupling rod of a locomotive 
which in addition to acting as a strut has to bear a lateral 
load due to its centrifugal acceleration. Each part of the 
rod moves in a circle of radius r, making n turns per second. 
The centrifugal force per unit length of the rod is therefore 

4 7T" 71? VTW 

in pounds weight, where ??i is the mass of unit length of 

the rod in lbs. 

The bending moment due to centrifugal force is greatest 
when the rod is at the top or bottom of its path : in each of 
these positions the effect is that of a lateral load equal to 

4*77"^ ??" ? 1 ??? 

per unit of length. When the rod is exerting longi- 

y 

tudinal thrust as well as running at a high speed the thrust and 
this lateral load should be taken account of jointly. In general 
however the bending due to centrifugal force is greatest under 
conditions which exclude longitudinal thrust, namely when the 
engine is running down hill with steam shut off. 



CHAPTER X. 



TORSION OF SHAFTS. 



127. Torsion of a uniform circular shaft. When a rod 
or shaft of uniform circular section is twisted, by applying opposite 
couples to its ends, the axis of the rod being the axis of the 
couples, the stress is everywhere one of pure shear. The strain 
may be regarded as due to a rotation of each plane of section 
relatively to neighbouring planes. At the centre the strain, and 
therefore the stress, is nil, and at other points of the section the 
amount of the shear is proportional to the distance from the axis. 
Assuming the strain to lie within the limit of elasticity the 
intensity of shearing stress q varies with the radius r. A radius 
DB (fig. 128) turns round to DC, and a straight line A B drawn 
parallel to the axis at any distance r changes into the helix AC. 




Fig. 128. 



Let <f> be the angle which this helix makes with lines parallel 
to the axis: then cf> is what we have called in § 19 the angle of 
shear, at the distance r from the axis; it is proportional to r. 
The angle 6 or BCD may bo called the angle <>t" bwist for tho 
length AB; it is proportional to AB. 



188 TOKSION OF SHAFTS. 

Taking two normal cross sections at a distance £x from one 
another we have 

C being the Modulus of Rigidity. 

The lines of principal stress are helices inclined at 45° to the 
direction of the axis. 

128. Relation of the greatest intensity of stress to the 
twisting moment in solid and hollow circular shafts. 

Since q varies as r we may write 

where q 1 is the intensity at the surface where it is greatest, and r a 
is the radius of the shaft. The whole moment of the shearing 
stress distributed over each cross-section must be equal to the 
moment applied by the twisting couple M. Over any ring of 
radius r and radial width Sr the intensity of stress is q and the 
total stress is q . ^irrhr. This acts at a radius r, and contributes to 
the whole moment the quantity 

q . 27rr 2 8r. 

Summing up these quantities for the successive rings into 
which the whole section may be conceived to be divided, we 
have 

M = fq.27rr 2 dr. 

Substituting — for q this gives 

7\ 

For a solid shaft the limits of integration are from r = to 

r = r, . 



Hence for a solid shaft 



2 ' 



TORSION OF SHAFTS. 189 

For a hollow shaft the internal radius of which is r 2 and the 
external r 1} 



= 7rg 1 (r 1 4 -r 2 5 

To express the greatest intensity of stress produced by a given 
twisting moment M we accordingly have 

2M 



2i = 



7T?Y 

when the shaft is solid, and 

2Mr 1 _ 

when it has a central hollow with radius r 2 . 

These results may be expressed for both cases by writing 

Mi\ 
qi = -jr, 

where J is the polar moment of inertia of the section, which in a 
hollow or solid round shaft has twice the value of / the moment 
of inertia about a diameter. 

It is important to notice how little the greatest intensity of 
stress is increased when a shaft is lightened by removing even a 
considerable portion of the mass from the centre. 

129. Angle of Twist in Round Shafts. Writing i for 
the angle of twist per unit of length we have 

._S<9_0_ 7l 
8x r C)\ 

whether the shaft is hollow or solid. This gives 

._M_ 
l ~~ CJ' 

. . 2M , 2M 

whence i= and —^7- .. 

for -the two cases. An application of this to tin- measurement of 
C has already been given in § 71. Another application La to 
observe the twist of a shaft as a humus of determining the couple, 
and from that the power, which the shaft Lb transmitting, 



190 TOESIOX OF SHAFTS. 

130. Relation of Power transmitted by a Shaft to 
Torsional Stress and Angle of Twist. In practical problems 
relating to shafting the data often are the speed of rotation of the 
shaft, and the number of horse-power it is to transmit. Let H be 
the horse-power and N the number of revolutions per minute. 
Then the work done per minute, in inch pounds, is 12 x 33000 H, 
and the angle turned through per minute is 2-rrN. Hence the 
twisting couple in inch pounds, 

12 x 330005 ™™# 

Applying this to a solid shaft of diameter d, 

__ 2M_ _ WM _ 3210005" 
qi ~ 7T7Y 5 " ~^¥ ~ Nd z ' 

W 



From this d — 68*5 * 

As a safe value of q 1 9000 lbs. per square inch is often taken in 
wrought-iron shafting, which makes 

* = 3 ' 29 \/jT 

for wrought-iron. 

The greatest stress q x may safely be 13500 for steel, and 4500 
for cast-iron. The corresponding expressions are 

3 /77 
d = 2-88 W ^ for steel ; 

^ = 4*15 a/ ^. for cast-iron. 

It has been assumed here that the twisting moment acting on 
the shaft is uniform. In many practical cases however the 
moment varies periodically. A shaft driven by a single crank and 
connecting rod, for instance, is subject to a moment which varies 
from a maximum to zero twice in each revolution. The propeller 
shaft of a steam-ship, driven by two, three, or more cranks suffers 
smaller, but still considerable, variations in twisting moment. 
When a fly-wheel intervenes between the source of power and 
the shaft it tends to smooth out such irregularities, but some 
irregularity in the moment remains, and in all cases where the 



TORSION OF SHAFTS. 



191 



moment is not uniform provision should be made by an appropriate 
increase in the value of d. 

Another reason for increasing d is present in most cases. The 
shaft is subject to a certain amount of bending as well as twisting. 
This arises partly from its own weight, partly from the weight of 
pulleys or spur wheels upon it, and partly from the lateral forces 
which are brought to bear on it by the gearing or belting through 
which it takes or gives off power. We shall see presently how to 
calculate the effect of a bending moment acting in conjunction 
with a twisting moment when the amount of the bending moment 
is known. But in many practical cases the bending moments to 
which the shaft may be liable can scarcely be specified with any 
certainty, and in such cases the practice is generally followed of 
increasing the diameter to provide for such contingencies, by an 
amount which experience of similar shafting has shown to be 
prudent*. 

131. Twisting combined with Bending. When a shaft 
is subjected to a known bending moment in addition to a known 
twisting moment we may apply the method exemplified in § 96 to 
find the magnitude and direction of the greatest principal stress. 
An important practical instance occurs in the case of a crank-shaft 
(fig. 129). Let a force P be applied to the crank-pin A at right 




Fig. 120. 

angles to the plane ABC. At any section G of the shaft, 
between the crank and the bearing, the force P gives rise not 
only to a twisting moment M t the amount <>t' which is r.All. hut 

* For practical rules on this point reference should be made to I'nwin's 
Elements of Machine "Design. 



192 TOESION OF SHAFTS. 

also to a bending moment M 2 the amount of which is P . BG. 
There is also a direct shearing force at the section C, the total 
amount of which is P, but this, as we have seen in § 95, is 
distributed over the section in such a way that its intensity is 
zero at the top and bottom of the section. It is at the top and 
bottom that the intensity of stress due to combined bending and 
twisting is greatest, and hence in calculating the greatest principal 
stress we have no direct shearing stress to take account of. At 
the top and bottom of the section there is, first, a normal longi- 
tudinal stress due to bending, the intensity of which is 

4Jf, 

P = — i > 

7T?Y 

and second, a shearing stress due to torsion, the intensity of 

which is 

2M, 

y = — »• 

When these are combined as in § 96 we obtain for the principal 
stresses the values 

Jp ± vV + ip 2 



_ 2 (M 2 ± Vi^ 2 + M?) 
irr x z 

Hence the greater principal stress has the same value as the 
stress which would be produced by the application of a bending 
moment of the magnitude 

without any twisting. This is sometimes called the equivalent 
bending moment. In the same way the quantity 



M 2 + Vil^ 2 + M* 

is sometimes called the equivalent twisting moment, as being 
the moment which if acting alone to produce torsion would 
produce a stress numerically equal to the greatest stress which 
the actual combination of bending and twisting moments pro- 
duces. The student should, however, be careful to notice that 
the greatest stress produced by the combination is a normal 
stress, whereas that due to a twisting moment is a shearing 
stress, and for this reason the conception of an equivalent bending 
moment is less open to criticism than that of an equivalent 
twisting moment. 



TORSION OF SHAFTS. 193 

Since M 1 = P . AB and M 2 = P . BC, the equivalent bending 
moment may be expressed as 

$P (BC + J AB* + BC*) 

or 

iP(BG + AC). 

The greatest shearing stress, due to the combined bending 
and twisting, being by § 96, \'q 2 + \pr, is equal to 



2v / i/ 1 * + i/ 2 2 _ 2P. AC 
rn\ z tti\ 3 

The axes of principal stress are inclined so that 

. M 1 AB 

^ n2e = Mr'BC' 

being their inclination to the section. 

In all cases of combined twisting and bending, whether the 
moments are due to forces applied to a crank or to other causes, 
the method here given may be applied to find the equivalent 
bending moment, or to calculate directly the principal stresses 
and the greatest shearing stress. The joint effect of the two 
moments is readily exhibited by drawing a diagram showing the 
equivalent bending moment at all sections of the shaft*. 

132. Resilience of a Round Shaft under torsion. The 

work done in twisting unit length of a shaft, within the limit of 
elasticity, is 

where M is the twisting moment and i is the angle of twist per 
unit of length. We have seen (§ 129) that 

9i 



l -c ri 



and Af = 



vr/V ry, 



2 ' 

Hence the work done per unit of length is 

* For examples see Unwiu's Elements of Machine Design, 

E. S. M. 13 



194 TORSION OF SHAFTS. 

and hence the mean resilience, per unit of volume of the 
material, is 

4C" 

an expression which may be compared with those already given 

for a rod under pull, and for a bent bar. 

If instead of a solid shaft we are dealing with a hollow shaft 
whose thickness is small compared with its radius, the resilience 
per unit of volume approximates to the value 

26" 

all the material being then subject to a stress which approximates 

to ( h . 

133. Torsion beyond the Elastic Limit. When the 
twisting of a round shaft is earned beyond the elastic limit the 
first portion to take permanent set is a ring round the circum- 
ference, and if the material is reasonably plastic the stress on this 
ring ceases to increase or increases onlv verv slightly when an 
increased twisting couple is applied. ^Yith increased torsion 
more and more rings become similarly affected, and the condition 
of the shaft ultimately approximates to one in which the shearing 
stress is uniform throughout the section. Thus if q be the 
ultimate shearing strength of the material, the twisting moment 
which is required to break the shaft approximates to the value 

for a solid shaft, or 

27rq (i\ 3 — r 2 3 ) 

3 

for a hollow shaft. 

The moment, in the case of a solid shaft, has therefore a value 
greater in the ratio of 4 to 3 than that which it would have if a 
uniformly varying distribution of stress were maintained. 

The ultimate strength of a shaft to resist torsion is not to be 
inferred from a knowledge of the shearing strength of the material 
any more than the ultimate strength of a beam to resist bending 
is to be inferred from a knowledge of the tensile strength and 
crushing strength of the material, and experiments on rupture by 



TORSION OF SHAFTS. 195 

torsion are not a satisfactory way of obtaining data with respect 
to shearing strength. 

134. Spiral Springs. An ordinary helical or " spiral " spring 
yields mainly by torsion. There is in strictness some bending as 
well, but when the slope of the helix is very small the bending is 
insignificant, and the strain may be treated as approximating 
closely to pure torsion. This is the case when the spring is closely 
wound and the diameter of the helix is large compared with the 
diameter of the wire or rod of which the spring is made. 

Let a be the radius of the helix and r the radius of the wire, 
which we assume to be circular in section. A load P, stretching 
the spring, exerts a twisting moment Pa on the wire. It produces 
a shearing stress, due to this torsion, the greatest value of 
which is 

2Pa 

The angle of twist per unit of length is 

q x 2Pa Pa 



% = 



Cr ttCV 4 GJ' 



The whole angle of twist is il y when I is the length of the wire 
composing the helix. It is this twist that causes one end of the 
spring to move out when the other end is held fixed. 

Every element in the length of the wire produces by its twist 
a displacement of the point from which the load is hung, through 
a distance equal to the product of the angle of twist into the 
radius of the coil. Hence the whole amount by which the spring 

is extended is 

.. 2Pa~l 
ail = n - . 

The work done in stretching the spring is half the product of 

this quantity into P, an expression which is easily shown to bo 

a 2 
equivalent to ^ n per unit of volume of the wire. 

A numerical comparison of the resilience of a spiral spring 
with that of a bent rod will show that a considerably larger amount 
of energy can be stored in a spring where the strain is torsional 
than in one where the material is strained by bending. 

13— 2 



196 



TOESIOX OF SHAFTS. 



The best disposition of all, that is to say the disposition which 
would allow a given weight of material to store the greatest 
amount of energy, would be that of a thin hollow tube of circular 
section, strained in torsion. A spiral spring made of such a tube 

would store an amount of energy approximating to ^ per unit of 

volume. 

We have already seen that a rod directly extended has a 

resilience equal to ^-= , and a bent rod (if rectangular in section) 

when subjected to a uniform bending moment has a resilience 

equal to £=. Since E is generally about f 0, these quantities are 

less than the resilience of the twisted tube in the ratio of 2-J and 
7J respectively, if we assume that equal intensities of normal 
stress and shearing stress are permissible. 

135. Helix in which the obliquity is considerable. 

The spiral spring dealt with in the preceding paragraph was 
supposed to have coils so flat that the strain could be treated 
as simply one of torsion. When the obliquity of the coils is 
considerable, the effect of bending has to be taken account of. 
Let a be the inclination of the wire to a plane perpendicular to 
the axis of the helix (fig. 130). Then the moment due to the 




Pa sin a 



Fig. 130. 

load, namely Pa, which acts about the axis OY may be resolved 

into two moments, 

Pa cos a and Pa sin a, 



TORSION OF SHAFTS. 197 

acting about OT and OS respectively, of which Pa cos a produces 
twisting about the axis of the wire OT, and Pa sin ol produces 
bending about the axis OS perpendicular to the axis of the wire. 

Hence the angle of twist per unit of length is 

Pa cos a 

and the angle of bending per unit of length is 

Pa sin a 

Now resolve each of these angles about the axes OX and OY 
to find the horizontal and vertical components of the angular 
displacement, per unit of length of the wire. Call the horizontal 

rh 

and vertical components of angular displacement j and -y respec- 
tively, I being the length of the wire. The horizontal angular 
displacement is reckoned as positive when it implies increase of 
curvature in the helix. 

The horizontal component of angular displacement, about OX, 

i ,, • , • Pa cos a . . . ... mi i • i 

due to the twist is — ~-= — .sin a ana is positive, lhe horizontal 

component of angular displacement due to the bending is negative 

, . . , . — Pa sin a 

and its value is — =-== . cos a. 

hi 



Hence 



e „ • (I i 

j = Pa sin a cos a I ~ T — -r=j 



Both bending and twisting produce positive vertical com- 
ponents of angular displacement, and the amount due to the 
two is 

<b „ /cos- a sin 2 a 

-r la {-cj + ei 

We are of course assuming here that the strains arc small and 
that the principle of superposition is applicable. 

Hence the axial extension of the spring, which is ncp, is 

.. .,, , COS 8 a sin- a 
e = Pa-l( aJ + m 



198 TORSION OF SHAFTS. 

and the whole angular displacement of the free end in the 
horizontal direction is 

6 = Pal sin a cos a I -~j — -=j J . 

In a wire of circular section J =21, and C is in general about 
| E. Hence the quantity 

CJ EI 

is in that case positive and its value is about ~rwj- The positive 

sign means that such a spring coils itself closer as it stretches*. 

When a is infinitesimally small 6 vanishes, and the expression 

Petri 
for the axial extension becomes -^-= as before. 

CJ 

The expressions given here for e and are not applicable to 
springs in which the section of the wire is other than round, for, 
as will be shown in the next paragraph, the torsional rigidity of 
other sections is not correctly, expressed by CJ. It is, however, in 
general not excessively different from CJ, and hence it will be 
obvious that if a section be chosen in which J is very much 
greater than I the value of 6 may come to be negative : in other 
words, a spring may be made which will unwind when it is 
stretched. Let the spring for instance be wound from a thin 
strip, so that the section has a depth much greater than its width 
measured radially to the coil. Then / is small and J is relatively 
very large. Although the expression given above for 6 is not 
applicable to such a section without modification, it serves to 
show that may be expected to be negative in such a case, and 
to have a comparatively large value. With such a spring there 
is, in fact, a large rotation of the free end. Since this rotation is 
proportional to the applied load it may be used, instead of the 
extension, as a means of measuring the load. This property of 
springs wound from flat strips has been discussed by Professors 
Ayrton and Perry f, who have also applied such springs to a 
number of uses in the design of instruments. 

* The treatment of spiral springs is easily extended to include cases in which 
a known couple is acting horizontally, to wind up the spring, in addition to, or in 
place of, an axial load, and also cases in which horizontal angular displacement is 
prevented from taking place. See Perry's Applied Mechanics, Chapter xxviii. 

t Proc. Boy. Soc. No. 230, 1884. 



TORSION OF SHAFTS. 



199 



136. Torsion of non-circular shafts. It is only in shafts 
of circular section that the shearing stress is uniform at all points 
equally distant from the axis, and varies uniformly along the 
radius. If the shaft has any other form, the stress due to a 
torsional couple is distributed over the section in a much less 
simple manner. 




Fig. 131. 



To see that this is the case, consider the stress at any point 
on the surface of a shaft in which the edge of the section is not 
perpendicular to the radius. A shearing stress at P perpendicular 
to the radius there, namely PA, may be resolved into shearing 
stresses PB along the edge of the section and PC perpendicular 
to the edge of the section. Each of these must be associated with 
equal shearing stress in a plane parallel to the axis of the shaft. 
Thus PC must be associated with a shearing stress parallel to the 
axis on a plane tangent to the surface of the shaft at P. But 
such a stress cannot exist unless forces parallel to the length of 
the shaft are applied at the boundary. In other words, a simple 
twisting couple cannot by itself produce in a non-circular shaft a 
distribution of stress such that the direction of shear is every- 
where perpendicular to the radius. To produce such a distribution 
would require the application of longitudinal forces to the boundary, 
in addition to the twisting couple. 

M 

It follows that we cannot apply the formula T to reckon the 

angle of twist in a square or, generally, a non-circular section, nor 

Mr 

treat the intensity of stress as — j , as it was in a hollow or solid 

shaft of circular section. 

The actual distribution of stress produced by Bimple twist in 



200 TOKSION OF SHAFTS. 

shafts of square, triangular and other forms of section has been 
investigated by St Venant, who has shown that in a shaft of 
square section the greatest intensity of stress occurs at the middle 

71 T 

of each side, and that its value then is r> _- f> 7o , h being the side 

0'208 th* 

of the square. This intensity is greater, by about one-seventh, 

than the greatest value which would have been found if the 

Mr 

formula — =- were applicable. Again, St Venant shows that the 
u 

torsional rigidity of a square shaft is 0*84 CJ instead of CJ; in 

other words, that the angle of twist of a square shaft per unit of 

length, due to a twisting couple M, is 

M 
084 CJ' 

The torsional rigidity of a square shaft is consequently less than 
that of a solid round shaft of the same sectional area in the ratio 
of 0-88 to 1. 

When the section is an equilateral triangle St Venant finds 
the torsional rigidity is 0'6CJ, which is 0'73 times that of a 
circular shaft of equal section. 

137. Stability of Shafts under End Thrust and 
Torsion. Professor Greenhill has investigated the influence 
which torsion produces on the stability of a shaft exposed to 
end thrust*. Taking end thrust alone the theory of Euler leads, 
as we have seen, to the conclusion that instability is produced by 
a thrust P when it is so related to the effective length L that 

ZL 2 _Z. 
L*~ EI' 

Professor Greenhill finds that when there is a torsional couple T 
acting on a round shaft in addition to end thrust, the condition 
producing instability is that 

7T 2 P T 2 



L 2 EI *EV 

The second term is so small, in practical cases, that it need not 
in general be taken account of in estimating the stability. 

* Proc. Inst. Meek. Eng., 1883. 



TORSION OF SHAFTS. 201 

The instability which may be produced in a shaft by the 
action of an end thrust is quite distinct from the instability 
described in the next paragraph. 

138. Centrifugal whirling of shafts. This action, although 
it has nothing to do with the torsion of shafts, may usefully be 
described when we are dealing with the causes of instability to 
which a shaft is liable. 

When a shaft revolves at a high speed its own inertia gives 
it a tendency to instability which is distinct from, although 
analogous to, the instability of a long column under end thrust. 
This instability is independent of any torsion to which the shaft 
may be subjected. It results from centrifugal force coming into 
play as soon as the shaft deviates from perfect straightness. At a 
particular speed the centrifugal force is just sufficient to keep the 
shaft bent and the state of things is then analogous to that of a 
column sustaining an end load equal to Euler's limiting value. 
The effect on the shaft is that when this critical speed is reached 
the amount of bending becomes large, for the centrifugal force 
increases pari passu with the deflection, and the shaft is then said 
to " whirl." 

Let w be the mass of the shaft per unit of length, and n the 
number of revolutions per second. Then the centrifugal force at 
any place where the deflection from straightness is u is 

4<7r-n 2 wii 

9 

per unit of length of the shaft. This is equivalent to a lateral 

load causing bending, and the bending moment M caused by it 

must be such that 

d 2 M _ 4<7r*nH vu 

da? g 

The curvature of the shaft due to this bending moment is 

</-//_ M 
dot? ~EI 

where / is the moment of inertia of the section about a diameter. 

Hence 

d 4 U ^7r-iriU(i 



202 TORSION OF SHAFTS. 

which for brevity we shall write 

d 4 u 
dx A 

, (kir 2 n 2 w\* 
using m to represent =j^ 1 . 

The general solution of this equation is 

u = A cosh mx + B sinh mx + G cos mx + D sin m#. 

When the shaft is simply supported at bearings but not held 
in them in such a manner as to fix its direction there, it is free to 
bend along the whole length L between the bearings. In that 
case the deflection and the bending moment are zero at each end. 

Hence, taking the origin at the middle, u = and -j— = when 

x = -=■ and when x = — -= . Further -7- = when x = 0. 

2 2 da? 

From these conditions it follows that A, B and D each =0, 
and the equation for the deflection becomes 

n= C cos mx, 

where G is the deflection at the middle. 



Then since u = when x = ^ 



from which 



2 

cos -s- = 0, 
2 



??liv 7T 



2 2" 

Hence the length L between bearings and the speed n which 
causes whirling are related to one another thus : 

T —- ( 9 EI V- ( 7r ' 2 9 EI \ z 
m \4)7r 2 n 2 wJ \ 4n 2 w / ' 

(gEI 

or 



tt /gEI 
= 2L 2 V io 



10 

When the' bearings are such as to fix the direction of the shaft 
at each end it may be shown that 

??iZ = 4'74. 

Other interesting cases arise when the shaft carries one or more 
pulleys, the mass of which has the effect of increasing the tendency 



TORSION OF SHAFTS. 203 

to whirl. The student is referred to a paper by Professor 
Dunkerley (Phil. Trans. Roy. Soc. 1894) where a large number of 
cases are considered in detail, and where an account will be found 
of experiments by which the results of the theoretical investigation 
were put to the test. 

The formula which expresses the condition giving rise to 
whirling in an unloaded shaft supported by bearings which do not 
fix the directions of its ends, namely 



7T gEI 

? *~22>V w 



may be put into a form more convenient for application to 
ordinary shafting. For iron or steel we may take E to be about 
30,000,000 lbs. per square inch, and w to be 0'28 lbs. per cubic inch. 
Hence for an iron or steel shaft the formula becomes 



it / 32-2 x 12 x 30,000,000 irr^ 
n ~2L*V ' 0-287rr 2 ' ' 4 ' 

where r is the radius of the shaft and L is the length between the 
bearings, both in inches. This gives 

160,000?- 

or Z = 400a/-, 

V n 

n being the number of revolutions per second as before. 



CHAPTER XI. 

SHELLS AND THICK CYLINDERS. 

139. Stress in a Thin Shell due to Internal Pressure. 

When a circular cylinder contains a fluid under pressure the 
material of the cylinder is thrown into a state of circumferential 
tension, which may be treated as sensibly uniform if the thickness 
of the wall is small in comparison with the diameter of the 
cylinder. Such a thin cylinder is called for brevity a shell, and 
the circumferential stress is called the hoop tension. The barrel 
of a cylindrical boiler is in effect such a shell : the thickness of 
the plates being so small relatively to the diameter that we may 
without sensible error consider the hoop tension to be uniform 
throughout the thickness of a plate. 

To find the relation of the hoop tension f to the radial pressure 
p, consider the equilibrium of a piece of a shell subtending a small 
angle 6 at the axis of the cylinder. The piece which is shown in 
elevation and plan in fig. 132 has four sides, two of which (AC 
and BD) are parallel to the axis of the cylinder, and the other 
two are perpendicular to these. As the piece is supposed to form 
part of a uniform circular cylinder there can be no shearing stress 
on any of the four sides, for the neighbouring pieces are under 
identical conditions, and further, whatever normal force acts on 
AB must be balanced by an equal and opposite normal force 
on CD. The only forces left to be considered are the hoop 
tensions on AC and BD, indicated by arrows in the figure, and 
the pressure of the fluid on the inner surface of the piece. 

The piece is in equilibrium under the action of the three 
forces P, T, and T, where P is the outward push which the fluid 



SHELLS AXD THICK CYLINDERS. 



205 



exerts on it, and T, T are the pulls exerted on its sides AC and 
BD by neighbouring pieces, in consequence of the hoop tension. 




-> 



Fig. 132. 

Let the width of the piece measured parallel to the axis, namely 
AC or BD, be unity. The circumferential length AB is rd. 
Then, since is small, P is sensibly equal to the product of the 
intensity of the fluid pressure into the area of the piece, 

P=pr6, 

and T=ft, 

t being the thickness of the shell. 

But P = TO 

by the triangle of forces (fig. 132). 

Hence ft=P r > 



/= 



_j)r 



t 



This result is evidently applicable to a thin shell exposed to 
external as well as internal pressure, if P be bakeD to represent 
the excess of the internal pressure over the external. 



206 



SHELLS AND THICK CYLINDERS. 



The same result can be arrived at almost more simply by 
considering the equilibrium of half the cylinder. Here P, the 




Fig. 133. 

resultant of the internal pressure, is 2prl, where I is the length of 

the piece under consideration. This piece is balanced by T + T. 

Hence 

2T=2prl, 

2ftl = 2prl, 



f= 



_pr 

T 



as before. 



140. Longitudinal Stress in Cylinder exposed to In- 
ternal Pressure. If the ends of the cylinder are held together 
by longitudinal stress on the cylinder itself, and not by separate 
stays or other supports, there will be a longitudinal stress, the 
amount of which is readily found by imagining a transverse divi- 
sion AB, fig. 134, and considering the equilibrium of either of the 



Fig. 134. 

two portions, say the portion to the right. The sketch shows the 
cylinder, with its end, in section. Whatever the form of the end 
be, the resultant of the internal pressure on it is a force P acting 
along the axis of the cylinder and equal to pS where S is the area 
of section of the shell, as a whole, namely the area 7rr 2 . This force 



SHELLS AND THICK CYLINDERS. 207 

is balanced by a longitudinal stress/ 7 acting over the ring which 

constitutes the section at AB, namely the ring whose area is 2irrt. 

Hence 

/' . 2irrt = p . 7TT 2 , 

r= pr 
J 2t' 

Thus the longitudinal tension /' is half the hoop tension/. 

It is clear that if the ends are held together by stays, or in 
any manner other than by the material of the shell itself, the 
relation stated here does not exist. It is easy to imagine cases in 
which the presence of stays prevents any longitudinal tension from 
coming on the shell, and even cases in which an excess of 
tightness in the stays may put the shell into a state of longi- 
tudinal compression. 

141. Spherical Shell. The tensile stress in a thin spherical 
shell is at once found by imagining a diametral plane of division, 
and considering the equilibrium of each half. The resultant fluid 

- pressure is p . ttt 1 : the resultant of the tensile stress on the ring 

section is/. 2irrt. Hence 

f== pr 

J 2t' 

142. Cylindrical Shell of oval section. In a cylindrical 
shell of oval section, fig. 135, such as the tube of 

a Bourdon pressure-gauge, the equilibrium of the 
halves separated by a plane AB shows that if / is 
the hoop tension at A or at B, 

2f.t=p.AB, 



p.AB 
J~~ 2 ' 

Similarly, if/' is the hoop tension at A' or B\ 




p.A'B' 
J 2 

Thus the greater hoop tension is found at the places of greater 
curvature. A piece of the shell at A or at /> is supported against 
the pressure within, not merely by the hoop tension which results 
from the curvature there, but also by shearing stress on the sides 
of the piece which face towards A* and />'. This shearing stress 



208 SHELLS AXD THICK CYLINDERS. 

is associated with a bending moment which acts upon any annular 
strip of the tube bounded by two parallel transverse sections. 
The bending moment varies, from point to point round the strip, 
both in amount and in sign, in such a way that its tendency is 
everywhere to make the section more nearly circular. 

143. Thick Circular Cylinder. We have next to consider 
the distribution of hoop tension in a circular cylinder so thick 
that the hoop tension cannot be treated as having the same 
intensity from inside to outside. 

We may regard the whole thickness as made up of a series of 
superposed rings. The radial pressure is transmitted from ring to 




Fig. 136. 

ring with reduced intensity, and the hoojD tension decreases as we 
pass out from one ring to the next. Consider a small piece of a 
ring, anywhere within the thickness. Let r be its inner radius 
and r + Br its outer radius. On the inner surface of such a ring 
there is a certain intensity of radial pressure p. We may write 
p + &p as the intensity of radial pressure on the outer surface, it 
being understood that hp will be negative in the usual case, 
namely when a cylinder has to bear an excess of internal pressure. 

The radial pressure p is one of the three principal stresses. 
Another is the hoop stress p', and the third is the longitudinal 
stress, parallel to the axis, which does not need to be taken 
account of in considering the equilibrium of the piece, since it 
has equal and opposite values on the front and back faces. 



SHELLS AND THICK CYLINDERS. 209 

Reckoning push stresses as positive, we shall find that p has a 
negative value when there is an excess of pressure within the 
cylinder. 

As before we take the width of the piece to be unity in the 
direction of the length of the cylinder, and assume 6 to be small. 

Then the whole radial force acting on the piece (reckoned as a 
force pushing it in) is 

(p + hp) (r + 8r) 6 — pr6, 

and this must be equal to the resultant of the forces arising from 
hoop stress, namely to 

p'Sr.0. 

Hence (p + Sp) (r -f- Sr) — pr— p'hr, 

or phr + r&p = p'hr, 

which may also be written 

A further relation between p and p' is got by considering the 
strains. Unless the cylinder be very short the strain must be of 
such a character that plane sections taken transverse to the 
length remain plane when strained : in other words the longi- 
tudinal strain is uniform. Assume further that the cylinder has 
free ends. Then the longitudinal strain is entirely due to the 
lateral action of the two stresses p and p' , and its amount is 

P , P' 
aE aE' 

Hence to make the longitudinal strain constant we must have 

p + p = 2a, 
where 2a is a constant. 

If we assume that the cylinder instead of having free ends 
is subjected to a uniformly distributed longitudinal stress p'\ the 
longitudinal strain is the sum of the direct effect of p" and the 
lateral effect of p and p', and in that case also the condition of 
constant longitudinal strain requires that the sum of p and p shall 
be constant. 

We have then the two equations 

/> + // as 2a, 

and pSr + r&p = p'Sr. 

K. s. M. I 1 



210 SHELLS AND THICK CYLINDERS. 

Substituting 2a — p for p' and multiplying by r, this second 

equation becomes 

r 2 Bp + 2prhr = 2ar8r, 

or ^(pr 2 ) = 2ar. 

Integrating, pr 2 = ar 2 4- b, 

b 

where b is another constant. 

Also p' = 2a — p = a — - . 

The constants a and b are to be found by considering the 
boundary conditions. 

In the ordinary case of a thick cylinder exposed to internal 
pressure only, let pi be the internal pressure and let the internal 
and external radii be r y and r 2 respectively. Then p — pi when 
r = r 1} p = when r = r 2 . 

Hence o = ^- 



.) > 



a = 



TV — Tj 



r 2 - — r^ 
Hence the hoop stress at any radius r is 



M 2 

I 2 



-piV^il + 7 
r r 2 _ r z 

the negative sign showing that it is a tension. 

The hoop tension has its greatest value at the inner surface, 
when r = r x . We there have 



?V — *i 

At the outer surface it is 



- 2p%r? 

r 2 2 - r-f ' 

Thus if f represent the greatest safe tensile stress which the 
material of the cylinder will stand, the greatest safe internal 
pressure is 

n 2 + i~i ' ■ 



SHELLS AND THICK CYLINDERS. 211 

It should be noticed that the hoop tension is always greater 
than the internal pressure, no matter how thick the cylinder is. 

We have assumed throughout that the stresses lie within the 
elastic limit and that the cylinder is free from initial internal 
stress. The formulas are applicable to the design of the cylinders 
of hydraulic presses and other thick tubes intended to resist 
internal pressure. 

Fig. 137 (p. 213) illustrates by the curve AB the variation of 
hoop tension in a tube whose external diameter is three times its 
internal diameter. It will be noticed that the external portion of 
the metal is under comparatively little hoop tension and con- 
sequently contributes comparatively little to the strength. 

144. Thick Cylinder exposed to External Pressure. If 

the exterior is exposed to a radial pressure p e and the inside is 
free from pressure, these boundary conditions give the following 
equations for the constants a and b, i\ and v* being the internal 
and external radii as before : 

, -p e rfr 2 2 
r 2 — 1 1 

a = 



fv* * , 'V* " 



Hence in that case the hoop stress at any radius r is 

Per*' ( 1 + 5) 



p = 



r* - r\ s 



As this has the same sign as p e , it represents a circumferential 
pressure. It increases towards the inner surface, and there, when 
r = r u it has the value 

2p e r.r 



r.r - ir 



145. Use of Initial Internal Stress in strengthening a 
thick tube. We may make the external portion of a thick tube 

more effective for the purpose of standing pressure from within it' 
we arrange that there shall bo a condition of initial Internal stress, 
of such a kind that the outer Layers are bending bo contract ever i he 

inner layers. This bhrOWS the inner layers into a State of initial 
hoop pressure, and when the pressure of (he fluid inside the tnhe 

1 I -2 



212 SHELLS AND THICK CYLINDERS. 

becomes operative, its first effect is to relieve them of this initial 
pressure and to throw additional tension on the external layers. 

One way in which a helpful condition of initial stress may be 
produced is to build up the tube out of two or more superposed 
layers, the outer being shrunk on over the inner. This device 
has been applied in the construction of big guns. The outer tube 
is bored to a diameter somewhat smaller than that of the barrel 
over which it is to be placed. It is then heated until it expands 
sufficiently to slip over the barrel, and as it cools its contraction 
makes it press on the barrel below, producing hoop tension in 
itself and hoop pressure in the barrel. Then when any pressure 
is applied inside the tube its effects in producing hoop tension 
are superposed on the existing hoop stresses, with the result that 
the outer portion of the compound tube becomes more severely 
stressed than it would be if no initial stress existed, and the inner 
portion is less severely stressed, since its hoop tension is reduced 
by the initial stress there. By this means the general distribution 
of stress due to internal pressure is considerably equalised. It is 
evident that this method of equalising the stress may be carried 
further by building up the whole tube in the form of a series of 
rings each shrunk on above those below it, and in gun-making it 
was at one time customary to shrink a series of tubes one above 
another. 

Confining our attention to the simple case where the cylinder 
is built up of two tubes, let p be the radial pressure at the 
surface between them, due to the shrinking on of the outer tube, 
apart from any stress that may be caused by application of 
pressure within the bore of the inner tube. Then the results 
arrived at in the last paragraph show that the condition of initial 
stress in the compound tube is as follows. In the inner portion, 
from radius r 3 to radius n, there is circumferential stress (of 
pressure) equal to 

f.*-,'(i+g) 

r 2 2 — Ti 2 
In the outer portion, whose radii are r 2 (internal) and r 3 (external) 
there is circumferential stress (of tension) equal to 



w(i+*i) 



r,- - r.S 



SHELLS AND THICK CYLINDERS. 



213 



Now suppose that the compound cylinder as a whole, with 
internal radius r x and external radius r 3 , is thrown into a state 
of stress by the application of an internal pressure pi. 

The hoop stress which this would cause at any radius r, 
apart from initial stresses, is 

-pir^il + 



r . 



The actual hoop tension is to be found by taking the sum of this 
and of the initial circumferential stress. 



CA 




Fig. 137. 



Fig. 137 illustrates this summation for a particular case. It is 
there assumed that r 3 = 3/ , 1 and ?\ 2 =2?- 1 . The inner tube has a 
bore of 2 inches radius ; its external radius is 4 inches, and on 
this an outer tube with an external radius of 6 inches is shrunk 
on. The dotted curves CD and EF represent the initial states 
of circumferential stress set up in the outer and inner tubes 
respectively by the shrinking on. The curve AB represents the 
hoop tension which would be caused in the tube as a whole, if it 
were free from internal stress, when an internal pressure is 
applied equal in this example to seven times the pressure al the 
surface of contact of the two tubes. The curves QH and JK 
represent the resulting actual state of hoop tension. There is of 
course a discontinuity in the hoop stress as we pass from one tube 
to the other. 



214 



SHELLS AND THICK CYLINDERS. 



The numerical results of this example are stated in the 
following tables. The radial pressure produced by shrinking 
on, p , is taken as unity, and pi=*7p . 

Circumferential Stress in Compound Cylinder. 





Radius 


Hoop 
pressure 
due to p 


Hoop 

tension 

due to p 


Hoop 

tension 
due to pi 


Eesultin: 

hoop 
tension 


Inner 
Tube 


( 2 


2-67 




8-75 


6-08 


il 


1-93 
1-67 




4-37 

2-84 


2-44 
1-17 


/^-.i J- n -» 


1° 

( 6 




2-60 


2-84 


5-44 


Uuter 
Tube 




1-95 


2-13 


4-08 




1-60 


1-75 


3-35 



The advantage of the initial internal stress is obvious when a 
comparison is made between the two last columns of the table. It 
reduces the greatest hoop tension from 8*75 to 6"08 by throwing 
a larger share of duty on the outer layer of material. By 
division of the tube into three or more parts, with appropriate 
pressure at each surface of division, a more equal distribution 
could be secured. In all cases the state of initial stress must of 
course satisfy the condition that the total hoop pressure and 
total hoop tension due to initial stress are equal. 

In the more modern method of gun-making a long strip of 
steel is coiled over a barrel to form the compound tube. By 
regulating the tension under which the steel strip is wound on the 
barrel a suitable condition of initial internal stress is produced*. 

146. Stress in a Revolving Ring. The problem of finding 
the circumferential stress in a revolving ring, due to the action of 
centrifugal force, is so closely analogous to the problem of the 
boiler shell that it will be convenient to mention it here. 

Consider a short portion of the ring, subtending a small angle 
at the centre. Let s be the area of section, p the density, and r 
the radius. When the ring is revolving with a velocity v, the 
centrifugal force on this piece, expressed in gravitational units, is 



psrO 



rg 



* The student is referred in this connection to articles by Prof. Greenhill in 
Nature, Vol. xlii. pp. 304, 331, 378. 



SHELLS AND THICK CYLINDERS. 215 

and this is balanced by the resultant of the two tensions at the 
extremities of the piece, namely 

fie 

where f is the intensity of the circumferential tension. Thus 

fse = psrW 

rg 

j. pv 2 
or J=—> 

9 

a result which shows that the hoop tension due to centrifugal 
force in a revolving ring depends only on the circumferential 
velocity and is independent of the radius of the ring. Thus in 
a running belt the tension, so far as it is due to centrifugal force, 
is independent of the size of the pulleys, and is the same in the 
straight and curved parts of the belt. In a steel ring, the density 
of which is 0*28 lbs. per cubic inch, a velocity of 567 feet per 
second produces a hoop tension of 15 tons per square inch. 

147. Stress in a Revolving Disc. A more difficult problem 
is presented by a revolving disc. The following solution is sub- 
stantially accurate when the disc is thin, and of uniform thickness. 
We have two principal stresses to deal with at any point, namely 
the radial stress p and the hoop stress p. We shall use the 
positive sign for tensile stress and shall express the stresses in 
absolute units to avoid the introduction of g into the equations. 

Let a) be the angular velocity of the disc, 
p its density, 

E Young's modulus, 

/jl Poisson's ratio, namely - , 

u the displacement of any point at radius r in the direction 
of the radius, due to the strain. 

Taking a small element of the disc with radii r and ?• + 6> and 
with radially directed sides, the equilibrium of the element 
requires that the centrifugal force, which is pco'-r per unit of 
volume, shall be balanced by the resultant of the hoop tensions 
p'on the sides of the element together with the difference of radial 

stress on the outer and inner faces of the element. This gives an 



216 SHELLS AND THICK CYLINDERS. 

equation between p and p' analogous to that in § 143 with the 
addition of a terra due to the centrifugal force, namely 

d 
P'^folP^ + rtpi* (!)■ 

Next consider the strains. The strain in the direction of the 

n . du 

hoop stress is - and the strain in the radial direction is -j- . Hence 
r r dr 

Eu , 

-~=P -MP- ( 2 )> 

and E-£=p-/j,p' (3). 

Combining (2) and (3) we have 

. E (U du\ JN 

p - T^Kr^H-) (4) ' 

E (llu du\ 
P = T^{r + ch-) ( 5 > 

Substituting these expressions for p and_p' in (1), 

dhl du U (1 — LL 2 ) (0*07* _ ,_ x 

r d? + dr—r + E =° ^ 

This gives, on integration, 

u_G (l-^ 2 )ft) 2 /?r 2 

r~? +Cl 8E (7) ' 

du C 3(1-^)0)^ 

3r*"JS + Ci SE (8) ' 

We have now to eliminate the constants by applying boundary 
conditions. Two cases have to be distinguished, namely the case 
of a complete or solid disc, and that of a disc with a hole in the 
centre. 

Taking first the disc with a hole in it, let r x be the radius of 
the disc and r the radius of the hole. Then p = when r = r 1 
and also when r = r . Applying these conditions in (4) and (5) 
we obtain 

^-^W/O^+r.' + ^-a + iWr*} (9), 



and 



p = °f(3 + n)(rs + r *- r ^-r^ (10). 



SHELLS AND THICK CYLINDERS. 21 7 

From (9) it follows that the hoop stress has its greatest value at 
the circumference of the hole, namely 

Greatest £>' = ^ {(3 + /i) n 2 + (I - fi) rA (11). 

And in the particular case where the hole is very small this 
becomes 

In the second case, that of a disc without a hole, the conditions 
are that p = when r = r 1 and u = when r — 0. 

Applying this latter condition in (7) we have = and 

M== <7 ir _5 8j£~ V 12 )' 

By substituting these values in (5) and applying the condition 
that p — when r = r x , 

Cl ~ SE (14) ' 

Hence, by (4), 

^ =ft ?{ (3+/i)n2_(1+3/tt)r 7 (15) ' 



»2. 



and j9 = W /(3 + /x)(r 1 2 -?- 2 ) (16). 

o 

Each of these stresses is a maximum at the centre, the maximum 
value being 

Hence the greatest hoop tension in a disc without a hole is just 
half that which exists when there is a small hole. 

These results show that if a thin disc of steel will bear safely a 
stress of 15 tons per square inch it may, when it has a hole in the 
centre, be whirled with a peripheral velocity of about 620 feet per 
second, and when there is no hole the speed may be increased to 
about 870 feet per second. It will be obvious that the strength of 
a disc to resist whirling is improved by giving it extra thickness 
in the neighbourhood of the nave, where the hoop tension is 
greatest. 



CHAPTER XII. 

HANGING CHAINS AND ARCHED RIBS. 

148. Loaded Chain. When a perfectly flexible chain or 
rope is acting as a suspension bridge to support weights, the 
condition that there can nowhere be any bending moment 
requires that at each point where weight is applied there must be 
equilibrium between three forces, namely the weight and the pull 
exerted by the chain on either side of the point. 





Fig. 138. 

Let ABODE be the chain carrying W 1} W 2) W s , TT 4 , as 
sketched in fig. 138. Then the pulls in the chain at A and B 
must equilibrate W lt and so on. Hence the form in which the 
chain hangs must be such that lines drawn from any point 
parallel to the successive sections of the chain will divide a 
vertical "line of loads" AE into segments AB, BC, &c. which 



HANGING CHAINS AND ARCHED RIBS. 



219 



represent the loads W 1} W 2 , &c. The lines OA, OB, &c. represent 
the pulls in corresponding portions of the chain. Further if a 
^ « horizontal line through be drawn to meet the line of loads in Q, 
AQ and J3Q represent the proportion in which the whole load is 
borne by the two piers, and OQ represents H, the horizontal 
component of the pull, which is constant throughout the chain. 

This construction allows us to determine a system of loads 
which will give an assigned form to the chain. The form is 
evidently unchanged when all the loads are altered in the same 
ratio — a process which is equivalent to moving the line of loads 
nearer to or further from 0. 

The converse problem is, given a system of loads, the distance 
of their lines of application from the piers being assigned, to find 
the form which will be taken by a chain carrying them. To solve 
this, find by the method of moments or otherwise the proportion 
of vertical load borne by each of the piers. Draw a line of loads 
AB, BG, &c. and divide it in this proportion in Q. From Q draw 
a horizontal line and take any point in it. Join with A, B, 
&c, and draw a chain, the successive sections of which are parallel 
to OA, OB, etc. 

Any chain drawn thus will be in equilibrium under the given 
system of loads, and to make the problem definite some further 
condition must be assigned, as for instance that the chain is to 
have a certain length, or that the horizontal pull between the 
piers is to have a certain value, in which case the length of QO 
in the force diagram is given. 

When load is continuously distributed, the chain forms a con- 
tinuous curve. The tension T at any point is equal to H sec i, 
where i is the inclination of the chain there to the horizontal. 
H is the tension at the lowest point. Any portion of the chain 




Pig. L89. 



220 



HANGING CHAINS AND ARCHED RIBS. 



is in equilibrium under the tangential pulls T lf T 2 at its ends and 
the resultant load R carried by that portion (fig. 139). Thus the 
tangents when produced meet in the line of action of R. 

149. Parabolic Chains. A case of considerable practical 
interest is that in which the chain bears a continuously distributed 
load which is uniform per foot run of the span. 

Taking the origin at the lowest point of the chain, the total 
load borne by any arc AB (fig. 140) is wAM = wx, w being the load 
per foot run of the span. This load acts through V, the middle 
point of AM. The tangent at B meets AM in V and consequently 
bisects AM. Hence 

y _ wx 



x 



tux' 



an equation which shows that the form of the chain is a parabola. 




Fig. 140. 

Writing c for the horizontal distance of the pier from A, and d 
for the depth of the vertex below the pier, we have 



H = 



1VX J w& 



2y 2d ' 
The tension at any point 

and at the ends, where the tension is greatest, this becomes 



w ° v5 + l 



HANGING CHAINS AND ARCHED RIBS. 221 

It is convenient to remember that the half length of a 
parabolic curve, from the vertex to either extremity, is approxi- 
mately 

2d 2 

and hence, if the half length undergoes a small change 8s, say 
through strain or through change of temperature, the amount by 
which the chain will sag is approximately 

ScSs 
If the piers are unequally high, we have distinct values for 

Ct\ . Cvo -— C\" . C>2 , 

(Ci -f c 2 ) \ld x L Vrfj 



c, = 



v 7 ^! + \ld. 2 ^d x + \/d. 2 ' 



an equation which serves to determine the position of the vertex 
when the span L and the heights of the piers are known. 

These results have application in the design of suspension 
bridges, where the total load supported by the chain, including its 
own weight, is nearly uniform per foot run of the span. Another 
application is to determine the stress in a telegraph wire, due to 
its own weight or to a load of snow. Strictly, in such a case the 
load is uniform per foot of the wire's length, but so long as the 
inclination is small there is no material difference between that 
and a load which is uniform per foot run of the span. The case 
of a load uniform per foot of the chain's length has little interest 
except as an exercise in applied mathematics. Its solution is 
given in the next article. 

150 Common Catenary : Uniform Chain loaded with 
its own weight. Here w is uniform per foot of Length, not per 
foot of span. On any arc s the load is ws. The horizontal 
component of tension H may be represented as tent where m is B 
certain length. Using i as before to express the inclination at any 

point, 

(hi . W8 S 

-£- = tan i = = — . 

(I.r win in 



222 HANGING CHAINS AND ARCHED RIBS. 



Then £V i+ © 2= ™ v ™ 2 +* 2 . 

Sx &s 

or — = / . 

in vm 2 + s 2 

rp 

Integrating,. — = \og e (s + Vm 2 + s 2 ) + G. 

lib 

Since 5 = when x = 0, C = — log e ??i, and 

a; , s + Vm 2 + s 2 
= log* , 

x 

or s + Vm 2 + s' 2 = ??ze'". 

m sinh 



Hence s = -^ \e m - e m ) = 



dy s . , x 

Also ~t- — — = sinh — , 

ax in in 

x 

from which y = ??i cosh in, 

J in 

the constant of integration being — m since y = when x = 0. 

This is the equation to the common catenary, taking the 
origin at the vertex. 

A neater expression is obtained by shifting the origin to a 

point at a distance m below the vertex. Then calling y the new 

ordinate, 

x 
v' = m cosh — . 
in 

The tension at any point is 

T = */H 2 + w 2 s? = w \/m 2 + s 2 



= turn * / 1 + sinh 2 — = wm cosh — 
v m m 

= wy'- 

That is to say, the tension at any point is equal to the weight of a 
chain whose length is the vertical distance of the point from a 
horizontal line drawn at a distance in below the vertex. 

Taking the origin again at the vertex, we have 

x 
y + in — m cosh — . 
° in 



HANGING CHAINS AND ARCHED RIBS. 223 

Also s 2 = m 2 sinh 2 — . 

m 

Hence 

(oc\ cc 

1 + sinh 2 — ) = m~ cosh 2 — = (y + m) 2 , 
mj m 

y = Vs 2 + m 2 — m, 

g 2 y% 

and m = — — - - , 

an equation which enables m to be found when the data are the 
length and the dip of the chain. When the data are the span and 
the dip, or the span and the length, the value of m may be found 
by trial from the equation given above. The following table gives 
the- ratio of m to the span corresponding to various dips, which 
are stated as fractions of the span : 

Dip — span g j^ j 2 j4 j-q Yg 2 0- 

?^span 1-023 T270 1*518 1766 2013 2261 2508. 

As an example, take the case of a wire stretched between level 
supports with a span of 1000 feet and a dip of 100 feet. The 
stress at the middle, or mw, is 1270w, w being the weight of the 
wire per foot. If the curve were treated as a parabola the 

calculated stress at the middle would be ,- (§ 149) or 1250w. 

This will show that even when the dip is as much as one-tenth 
of the span no material error results from treating the distribution 
of weight as uniform per foot run of the span. 

151. Suspension Bridge with Stiffening Girder. The 

flexibility of the hanging chain causes it to change its shape 
under moving loads and this is a serious drawback to the use of 
suspension bridges. If the platform is also flexible it may be 
thrown into a state of dangerous oscillation, especially when 
variations of load are repeated periodically, as happens for example 
when troops are marching over the bridge. The structure ni.w 
however be stiffened by the use of auxiliary girders tor that pur- 
pose. These girders are usually arranged in the manner indicated 
in fig. 141. Each chain carries a pair of stiffening girders, the 
length of which is equal to half the span, They are hinged to 

One another by a, pin at the middle M t and they rest on the pieifi 



224 HANGING CHAINS AND ARCHED RIBS. 

by pins and Q which are allowed sufficient freedom to slide in 




the direction of the span, and are held from rising as well as from 
falling, because with some distributions of load they may tend to 



rise. 



A single stiffening girder without a central hinge has also 
been used, but this has the disadvantage that the stresses on it 
depend on the initial tightness of the chain, and are much affected 
if through changes of temperature or any other reason the chain, 
becomes tighter or slacker. With a central joint, the stress in 
each part of the structure is at once determinate, and does not 
change, except to a trifling extent, when the relative lengths of 
the various pieces are altered by changes of temperature or other 
causes. 

To investigate the distribution of the load between the chain 
itself and the stiffening girder, when the girder is hinged, we may 
proceed as follows. 

The forces on each girder are the loads it carries, acting 
downwards, the pulls of the suspension rods, acting upwards, and 
the forces on its ends at the pins acting either downwards or 
upwards. These forces produce, in general, bending moment at 
any section of the girder. The bending moment on the jointed 
girders is zero at the ends and at the central hinge. 

We have seen in § 92 that the curve of a hanging chain is a 
diagram of bending moments for the load carried by the chain. 
Applying that principle to the chain with a stiffened girder, it is 
evident that the pulls of the suspension rods must always adjust 
themselves in such proportions, no matter how the load varies, as 
to make the curve of the chain represent a bending moment diagram 
for them, considered separately. In other words, out of the whole 
load carried by the structure, the chain carries directly, distributed 
through the suspension rods, as much as will constitute a system 



HANGING CHAINS AND ARCHED RIBS. 



225 



having a bending moment diagram of the shape of the chain. 
Hence if we draw a diagram of bending moments for the complete 
system of loads which the structure carries as a whole, and subtract 
from that a diagram of bending moments having the shape of the 
chain (using the appropriate scale so that this latter diagram will 
represent the bending moments due to the pulls of the suspension 
rods), then the difference between the two diagrams will show the 
bending moments which have to be borne by the stiffening girders. 

The diagrams may be drawn superposed, so that their differences 
are found graphically. The scale on which the curve of the chain 
represents a diagram of bending moments is to be determined 
with reference to the consideration that the resulting moment on 
the girder is zero at the central hinge. Consequently the total 
load diagram of moments must intersect the chain -load diagram 
of moments (that is, the chain curve) at the point which corre- 
sponds to the position of the hinge. 

An example will make this construction more intelligible. 
Let the curve represent the given form of the chain, which is 
drawn inverted so that it may resemble the bending moment 
diagrams with which the student is familiar. This curve represents, 
on a certain scale which we do not at first know, a diagram of 




bending moments for what has been called here the chain-load, 
that is to say, that part of the load which the chaiu carries in 
consequence of the pulls in the suspension rods. Now lei a 
bending moment diagram be drawn for the whole load carried by 
the structure. Draw that on the same base, selecting for it Buch 
a scale as will make that diagram intersect the other at the place 
where the hinge comes in the jointed girder (namely, at the 
middle of the span). The scale thus chosen is the scale on which 
both diagrams are to be interpreted. The difference between the 
two diagrams gives, on the same scale, the bending moments on 

E. S. M. 1 B 



226 HANGING CHAINS AND ARCHED RIBS. 

the jointed girders. Thus, in fig. 142, let the load. consist simply 

of a single concentrated load applied at a point P distant a from 

one end of the span, and b from the other. The diagram of such a 

load is a triangle, with the vertex over P, and we have to draw 

this triangle so that the side DQ shall intersect the chain curve 

at C, the place corresponding to the hinge. Hence the diagram 

is to be drawn by joining QC, and producing it to meet a vertical 

line through P in D, and then joining OD. The value of DP as 

Wab 
a bending moment is known, being —y— where L is the span. 

This determines the scale of moments for the figure. The over- 
lapping pieces ODC and CFQ are then the bending moment 
diagrams of the two girders. The right-hand girder in this case 
bears negative moments : that is to say it is bent up by the pull 
of the suspension rods on it, and is held down at its ends by the 
pins there. Having in this way determined the bending moments 

on the girders we may go on to find the shearing forces f -=— 1 . 

The shearing forces at the ends and middle of the span are the 
forces which have to be borne by the pins. 

This construction is applicable to any form of chain curve, and 
may be used to determine the bending moments and shearing 
forces which have to be provided for under any assumed distribution 
of load, and also the pulls in the suspension rods. If we assume 
that the form of the chain is a parabola, the pulls in the suspension 
rods are equivalent to a uniformly distributed load, and the case 
represented in fig. 142 may be expressed algebraically thus : 

Let w represent the load per foot run which is equivalent to 
the pulls of the suspension rods. Then CM represents, in the 

diagram of bending moments,, a moment equal to —~- . The load 

Wab 
W applied at P produces there a moment DP equal to -=- , and 

Wa 

at the middle a moment CM equal to 9 - . Hence, since the joint 

is at the middle, 

wL 2 _ Wa 

~~8~ ~ 2 ' 
4TFa 



HANGING CHAINS AND ARCHED RIBS. 



227 



which determines the amount of pull in the suspension rods when 
the number of the rods is known. The greatest bending moment 
M p to be borne by the left-hand girder, for this position of the 
load, is DE, or DP - EP. 

But DP = -^ and EP = — - (L - a), and substituting 

L 2 v J ° 



4 W a 



for w we obtain, for the bending moment at P on the 



stiffening girder, 



, a _ 3a? 2a s 

Mp *y^\ L L2 + D 



To find the position of the load which will make this moment 
dM 
da 



a maxir i we write - - = 0, which gives 



a = ^(l± ^=0-21lZ, or 0-789Z. 

The maximum moment to be borne by the stiffening girder is then 
readily found to be 

0-096 WL* 

This is the maximum positive bending moment which each 

girder will have to bear when a load W moves across the bridge. 

The maximum negative moment occurs at the middle of each 

girder when the load is at the middle of the span. Its value 

(fig. 143) is 

FG = FN-GN=\CM 

= tV WL. 




The student will find it instructive to extend this method of 
determining the moments on the stiffening girders to the case 
of a uniform advancing load. 

* This quantity is only very little greater than the moment produoed by placing 

W in the middle of the half girder, namely ..■. WL. 

15—2 



228 



HANGING CHAINS AND ARCHED RIBS. 



152. Inverted Chain. The Arch. Imagine a chain 
consisting of a series of stiff jointed rods, loaded at the joints, 
to be inverted so that it forms an arch, the form of the chain 
and the distribution of loads remaining unchanged. The chain 
is still in equilibrium, but its equilibrium is now unstable. Each 




Fig. 144. 




Fig. 145. 




Fig. 146. 
rod is now exerting thrust, instead of pull, and at each joint the 
thrusts of the two rods meeting there equilibrate the load applied 



HANGING CHAINS AND ARCHED RIBS. 229 

at the joint. But any variation in the distribution of load or 
any casual disturbance of the form of the chain upsets this 
balance and the arch yields. 

Next suppose that in place of being made up of jointed rods 
the arch is composed of blocks or " voussoirs " as in the model 
shown in fig. 144. In that model the voussoirs have slightly 
curved surfaces which allow them to rock to some extent on 
one another, thereby exaggerating the action which the elastic 
compressibility of the material permits in an arch of stone blocks 
with flat faces. It is clear that this rocking action gives the arch 
a stability which was not possessed by the inverted chain of rods. 
The loads may alter, or the form of the arch may be disturbed, 
to a limited extent, without causing the structure to break down. 
Each voussoir remains in equilibrium provided that the vertical 
force made up of its own weight and any load applied to it can 
be balanced by the two forces which are exerted upon by its 
neighbours through the joints between it and them. A con- 
tinuous line may be drawn to represent the direction of the 
thrust at all the joints as in fig. 144. This is called the linear 
arch : it represents the form of an inverted chain carrying the 
same system of loads. 

When the distribution of load is altered, the voussoirs turn 
slightly with the effect that the linear arch takes a form which is 
proper for the new distribution. In fig. 144 we have, besides the 
general load due to the weight of the voussoirs, an extra load at 
the crown, and consequently the linear arch rises there, just as a 
chain already in equilibrium under a distributed load would sag- 
more considerably wherever an extra load was applied. Similarly 
in fig. 145 the linear arch rises at the haunches to meet extra 
loads applied there, and is fiat at the crown, while in fig. 146 it rises 
to meet the load on one haunch. Precisely similar changes occur 
in the form and position of the linear arch in a ring of stone 
voussoirs, although there the absence of curvature at the joints 
prevents the action from being visible as it is in bhe model with 
curved blocks*. When the linear arch deviates from the middle 
of any joint, the resultant stress is no Longer axial, and the 
varying distribution of stress on the surface then causes a varying 

Figs, 114-110 arc due to Eleeming Jenkin, and are oopied from lii* artiole 
"BridgcH" in the Encyclopaedia Britdnnica, 



230 HANGING CHAINS AND ARCHED RIBS. 

amount of elastic compression which implies, in effect, some 
turning on the part of each voussoir. 

The limits of loading within which the structure remains 
stable are determined by the consideration that the linear arch 
may not pass outside of any joint. If, for instance, in fig. 146 
the load on the left were increased, one of the joints between the 
crown and the right-hand abutment would open because the linear 
arch would reach the inner extremity of the joint. Besides this 
condition, it is evidently necessary that the blocks should not slip 
on one another at the joints : in other words the direction of the 
linear arch at each joint must not make with the normal an angle 
greater than the angle of repose. In the model the joints are 
roughened to allow the linear arch to be largely displaced without 
causing slip. 

For reasons which have been explained in § 82, the rule is 
generally followed in the design of large stone arches that the 
linear arch be not required to pass outside the middle third of each 
joint. When this requirement is satisfied the effect is that all 
the surface at each joint is in compression, with an intensity 
which may range from zero at one edge to a maximum at the other. 
An obvious further practical condition is that the joints should 
be wide enough, in relation to the load, to prevent the greatest 
intensity of this compressive stress from exceeding a safe value. 

153. Arched Rib. A stiff rib, say of metal, shaped in the 
form of an arch, differs from a ring of individual voussoirs in this 
respect, that the linear arch need not lie within the section of 
the rib. In other words, the resultant of the stress at any section 
of the rib may lie outside of the section, because the rib is 
capable of sustaining bending moments. When the position of 
the linear arch is known, and the amount of the thrust in it, it is 
easy to find the bending moment which acts at any section of the 
rib. Each section has, in general, to bear bending moment, direct 
thrust, and shearing stress. We shall consider a rib carrying 
vertical loads. 

Let AB (fig. 147) be a section of the rib and let DE be drawn 
tangent to the linear arch at the same place. The resultant 
thrust F has the direction and position indicated by DE. This is 
equivalent to a parallel force applied at C, the centre of gravity 
of the section, together with a bending moment F.CE, if CE 



HANGING CHAIXS AXD ARCHED RIBS. 231 

be drawn perpendicular to BE. The force F acting through C 





Fig. 147. 

may be resolved parallel and perpendicular to the section. The 
component parallel to the section causes shearing stress : the 
component perpendicular to the section causes a uniformly dis- 

F 

tributed stress of compression -^ (S being the sectional area of 

AB), which has to be superposed on the stress due to the bending 
moment in finding the whole normal stress at any point of the 
section. 

Now the thrust F at any point in the linear arch may be 
resolved into a vertical component V and a horizontal component 
H, of which H is constant for all points in the arch. Comparing 
the triangles FVH and DCE of Fig. 147 we have 

H:F=CE: CD. 

Hence the bending moment, which is F.GE, is equal to H.CD. 
In other words, since H is constant, the height of the linear arch 
above or below the centre line of the rib constitutes a diagram of 
bending moments for the rib. The linear arch itself, as we have 
already seen in dealing with hanging chains, takes a form which 
is that of a diagram of the bending moments which the same 
system of loads would produce on a straight beam. Hence the 
problem of drawing the linear arch for the rib resolves itself into 
drawing a diagram of bending moments for a similarly loaded 
beam, with an appropriate scale and subject to the particular 
terminal conditions which apply in any given case. 

154. Rib hinged at the ends and centre. We shall first 

take the comparatively simple case in which the rib is hinged at 
the ends and also at the crown (or to put it more generally) at 
one other point besides the ends (tig. 148). Wherever there i- a 



232 



HANGING CHAINS AND ARCHED RIBS. 



hinge the bending moment on the rib must be zero, if we leave 
out of account the trifling amount of bending which a hinge may 







Fig. 148. 

take in consequence of friction. Consequently the linear arch 
must pass through the centre of each hinge. We proceed to draw 
a diagram of bending moments for the given loads, considered as 
acting on a beam of the span OQ. If this diagram passed through 
the third hinge it would be the true linear arch, for in that case 
the condition would be satisfied that there is to be no bending 
moment on the rib at P, as well as at and Q. In other words 
we have to select such a scale for the bending moment diagram, 
drawn on the base OQ, as will make it pass through P. This is 
readily done by first drawing it to any scale, say OKQ, and then 

PM 

reducing all the ordinates in the ratio fryr* 

The linear arch OJPQ having been found in this way, the 
distance JC between it and the central curve of the rib gives, 
on the same scale, the bending moment which has to be taken 
by the rib. The amount of the thrust F at any place is also 
determined, like the pull in a hanging chain, from the known 
form of the linear arch and the known values of the loads. Hence 
the stress at any section of the rib is found. 

It will be clear that the construction by which the linear arch 
is found applies whether the loads are symmetrically or unsym- 
metrically distributed. The student will notice the correspondence 
between the problem of the hinged arch and that of the chain with 
hinged stiffening girder/treated in § 151. 



HANGING CHAINS AND ARCHED RIBS. 



233 



155. Rib hinged at the ends only. In the case discussed 
in the last paragraph, namely that of a rib hinged at three places, 
the problem of finding the linear arch is perfectly determinate : 
the jointed rib cannot be self-strained as a whole, and with a given 
system of loads only one distribution of stress is possible. It is 
not affected by any yielding of the abutments, or by expansion 
or contraction of the rib through changes of temperature. But 
if the rib is hinged at the ends only it is clear that this is no 
longer true. The rib is bent if the span increases by any yielding 
of the abutments, or if, while the span remains fixed, the rib itself 
expands through heat. Hence it may be subject to bending 
moments apart from the effects of applied load. To make the 
problem of finding the linear arch determinate, we must introduce 
some assumption as to the initial state of stress before loads are 
applied and as to the fixity of the abutments. In what follows it 
will be assumed that the span does not alter, and that no stress 
exists in the rib except what is caused by applied loads. 

With these assumptions the problem is determinate, and the 
linear arch is to be found from the consideration that the total 
effect of the elastic bendings which the loads cause in every part 
of the rib is such as to produce no change of span. In other 
words, that 

2&» = 

where Sx is the horizontal displacement of one end. of the rib, 
relatively to the other end, which results from the bending 
of any element of the rib, the sum of such displacements being 
taken for all the elements. The linear arch must of course have 
its extremities in the hinged ends, as in previous cases, and it 
must be a diagram of bending moments for a similarly loaded 




Fig. 1.49. 



234 



HANGING CHAINS AND ARCHED RIBS. 



beam. The further consideration, that the bending must be such 
that X8x = suffices to fix the height of the linear arch. 

In order to consider the horizontal displacement of due to 
any one element 8s, taken alone, we must think of the portions of 
the arch lying to either side of the element as behaving for the 
moment like rigid bodies. Then if the position of the opposite 
end be considered fixed, the elastic bending of 8s would cause 
to move through a small distance Oa perpendicular to CO. The 
horizontal component of this displacement, or 8%, due to the 
bending of 8s, is Ob. Thus 

8x__CN = jy_ 
Oa~~CO~ CO 

where y represents the ordinate CN of the centre line of the rib. 

Let M represent the bending moment acting on 8s, then the 
angle through which the sides of 8s turn relatively to one another 
in consequence of bending is 



M 



8s. 



Hence 



0a = 



EI 
M.C0.8s 



and 



8x = 



EI 

My8s 
~EI 



Hence the condition that the span is not to be altered by 
loading makes 

2^ = 0. 




HANGING CHAIXS AXD ARCHED RIBS. 235 

It should be added that we are here neglecting the direct 
effect of the compressive stress in each element in altering the 
span, an effect which is very small compared with that of bending. 

Now the linear arch OJQ (fig. 150) is related to the centre line 
of the rib OCQ in such a manner that CJ is everywhere propor- 
tional to the bending moment on the rib. The condition that 
%Bx = may therefore be expressed 

2 CJ i^ = o. 

Suppose that the summation is made for a series of points 
taken at equal distances along the rib ; we then have 

But GJ=JN—y, and the condition becomes 

If a bending moment diagram OKQ for the beam carrying the 
given loads be drawn to any scale, its ordinate bears a constant 
ratio to those of the required linear arch, so that we may write 

JN = rKN, 
and 2f = 2^ Y =^f\ 



Hence 



r = 



2 yKN ' 



a quantity which is readily calculated after measuring CN and 
KN at a series of equidistant sections. Thus when r is found the 
diagram OKQ is transformed into the required linear arch by 
changing its vertical scale in the ratio r : 1. 

The procedure therefore is, to draw the curve of the rib, OCQ, 
and also on the same base a diagram OKQ of bending moments 
for the similarly loaded beam. Then taking a scries of sections 
at equal distances along the arc of the rib measure CN and K y 
and calculate r, which gives the ratio in which the scale ^( the 
bending moment diagram is to bo altered in order to give the 
linear arch. Then draw a second bending moment diagram OJQ 
to this altered scale. The distance CJ at am section measun - - i 



236 HANGING CHAINS AND ARCHED RIBS. 

that scale the bending moment which acts on the rib. The 
construction applies to unsym metrical as well as to symmetrical 
loading. 

156. Rib fixed at the ends. Here again we have to make 
certain assumptions without which the problem of finding the stress 
is indeterminate. We may assume as before that there is no 
initial stress, that is no stress except what is due to the loads, and 
that the abutments do not yield. The given condition that the 
ends are held fixed implies, in general, that when the rib is loaded 
there is some bending moment at each end. We shall take the 
rib to be symmetrical and symmetrically loaded, in which case the 
moment is the same at each end. Call it fi. Then the linear arch 
is a diagram of bending moments made up of (1) the moments in 
a beam carrying the same loads, and (2) the moment fi superposed 
on that : in other words the ordinate of the linear arch is 

JN - /x, 

where JN is, as in fig. 150, the bending moment on a beam 
carrying the same system of loads. The end moment is negative 
because it tends to reduce the amount of bending; such a beam 
would experience. Thus the linear arch retains the form which it 
has in a rib hinged at the ends, and is simply shifted downwards, 
parallel to itself, through a certain distance /x, which we have to 
find. Hence at any section of the rib the actual bending moment 
is now 21 — /x, where M is, as in § 155, the moment in a rib with 
hinged ends. The condition that 18x shall be zero still applies, 
and now leads to the result 

K (M-p)ybs 

EI ' 

% (CJ-r)yhs^ 

where J represents as before a point on what would be the 
linear arch for a rib with hinged ends. Since CJ = JX — y and 
.AY = rK2F this becomes 

r ^KX_st_^l i = (i), 

summation being made at a series of equidistant sections along 
the rib from end to end. 



HANGING CHAINS AND ARCHED RIBS. 237 

The further condition that the inclination of the rib at the 
ends remains unchanged gives another equation involving the 
two unknown quantities r and /x. Taking any element 8s along 
the rib, the bending in it considered alone would change the 
inclination of its ends, relatively to one another, by the amount 

(M - p) 



EI 



8s, 



and hence 2 - pj ^ s = 0> 

or 2, j .— — = 

for a series of equidistant sections. 

If we substitute rKN — y for CJ this gives 

r2~-2|-/^ = (2). 

The equations (1) and (2) together enable the quantities r and fi 
to be found. 

If the loading is unsymmetrical we have different moments /^ 
and yu 2 at the left- and right-hand ends respectively, and at any 
intermediate point the true moment acting on the rib is 

M-fi 

CO 

where fi = y^ — j (/*i — fa)> 

L being the span and x the horizontal distance from the left-hand 
end. This quantity is to be introduced into the equation stated 
above, and a third relation is found by equating the sum of the 
vertical displacements of one end, or 2S?/, to zero, a condition 
which holds when (as here) the opposite end has its direction as 
well as its position fixed. 



APPENDIX. 

The following Tables contain a few representative data 
regarding the Strength and Elasticity of Materials. 



/. Strength to resist Tension. 

Wrought-iron : — 

Finest Lowmoor and Yorkshire plates, tested in direction 

of rolling 
Finest Lowmoor and Yorkshire plates, tested across 

direction of rolling 
Staffordshire plates, in direction of rolling 

„ „ across direction of rolling ... 

Average good boiler plates, in direction of rolling 

,, ,, ,, „ across direction of rolling 

Ship plates, in direction of rolling ... 
„ „ across direction of rolling 

Finest Lowmoor and Yorkshire bars 
Average good bars 



Soft Swedish bars 
Charcoal-iron wire, hard drawn 
annealed... 



abo 



it 



Steel : — 

Ordinary mild steel bars and plates with about 0'2 per cent 

of carbon 
Specially mild steel 

Steel for rails, with about 0*4 per cent, of carbon 
High carbon steel for springs, annealed 

„ „ „ tempered 

Steel castings ... 

„ „ annealed 

Steel wire, ordinary ... ... ... ... ... aboi 

,, ,, tempered ... 



Ton 

squar 


s per 
e inch. 


27 


to 

24 
26 

24 
25 
20 


29 


20 


to 
19 


24 


24 


to 
25 
20 


29 


35 


to 
30 


40 


28 


to 


32 


2 i 


to 


36 


36 


to 


i:> 


if) 


to 


B0 


60 


to 


70 


L5 


to 


45 


25 


to 
70 
L00 





240 



APPENDIX. 



Steel — continued : 

Pianoforte steel wire 
Xickel steel with about 5 pei 

12 

Chrorne steel 
Tungsten steel .. 

Cast-iron 

,5 5, 



cent, of nickel, annealed .. 



average about 



sphorus 



Copper, cast ... 

„ rolled or forged 

„ wire, annealed 

,, „ hard drawn 

Copper with 0"2 to 0'4 per cent, of phc 

Ordinary yellow brass, cast '66 per cent, copper, 34 per 

cent. zinc- 
Ordinary yellow brass, rolled 

Brass wire 

German-silver wire ... 

Gun-nietal (about 90 per cent, copper and 10 per cent, tin) 

Phosphor bronze 

„ „ wire, hard drawn 

Aluminium, cast 

„ rolled ... 

Aluminium bronze (90 per cent, copper, 10 per cent, aluminiiun 
Zinc, cast ... ... ■ ... 

„ rolled 

Lead about 

Tin 

Soft solder 

Timber tested in the direction of the fibre :— 
Oak 

White pine 
Pitch pine 
Eiga fir ... 
Ash 
Beech 
Teak 
Spanish mahogany 

Cement, set for 1 week 
„ „ 1 year 

Leather belting- 
Hemp rope 



Tons per 
square inch. 

120 to 150 
40 
90 
80 



5 to 15 

8 

8 to 12 

13 to 16 

18 to 20 

26 to 30 

20 to 22 

10 to 12 

15 to 24 
20 to 25 

30 

12 to 17 

16 to 18 
45 to 70 

4 to 6 

6 to 10 
40 

1 to 3 

7 to 10 
1 

1 to 2| 
3 



3 to 7 
1| to 3^ 

4 

2| to .H 

4 to 7 
4 to 6 
4 to 7 
4 to 7 

016 

0-24 

■2 

4 to 5 



APPENDIX. 



241 



II. Strength to resist Crushing. 



Wrought-iron 
Cast-iron 

„ „ average 

High-carbon steel, hardened by quenching 
Brass ... 

Timber 

Cement 

„ concrete 
Portland stone 
Sandstone 
Yorkshire grit 

Slate 

Basalt ... 

Granite 

Brick, London Stock 

„ Staffordshire blue 

Glass 



about 



Tons per 
square inch. 


16 


to 


20 


25 


to 


65 


40 


to 


50 


120 


to 
5 


180 


2 


to 


4 


1* 


to 

1 

2 


2 


2 


to 
3 


5 


5 


to 


10 


8 


to 


10 


6 


to 


10 


| 


to 


li 


2 


to 


6 


10 


to 


15 



777. Strength to resist Shearing. 

Wrought-iron bars, across the direction of rolling 
,, ,, plates, ,, ,, „ ,, 

„ „ • „ in the plane of rolling ... 

Mild steel 

Cast-iron 

Timber (along the fibre) 



18 to 22 

16 to 20 

8 to 12 

21 to 25 

6 to 13 

i to 1 



IV. 



Wrought-iron 
Steel ... 
Cast-iron 
Copper, cast 

„ rolled 
Brass . . . 
Bronze ... 
Gun-metal 
Silver ... 
Gold ... 
Platinum 
Phosphor bronze 
Aluminium bronze 
Timber 

E. S. M. 



Moduluses of Elasticity. 

E 

Tons per 
square inch. 

12500 to 13500 

13000 to 14000 

4500 to 7000 

5000 to 6000 

5500 to 7500 

5000 to 6500 

6000 to 7000 

5000 

4800 

5400 

10500 

6000 

6500 

600 to 950 



C 

Tons per 

square inch. 

5000 to 5500 

5200 to 5700 

1700 to 2700 

1900 to 2300 

2100 to 2900 

2000 to 2300 

2300 to 2700 

1900 

L800 

2100 

4000 

28 K I 

2500 

16 



242 



APPENDIX. 



V. Approximate Weights of Materials. 





Lbs. per 




cubic foot. 


Metals : — 




Wrought-iron 


480 


Steel 


490 


Cast-iron 


430 to 470 


„ „ grey foundry ... 


450 


Copper 


550 


Brass, cast ... 


520 


„ rolled or drawn ... 


530 


Gun-metal 


540 


Aluminium, pure ... 


162 


„ commercial ... 


165 to 170 


Zinc ... 


450 


Tin 


465 


Silver 


655 


Lead 


710 


Gold 


1200 


Platinum 


1340 


Timber : — 




Oak 


50 to 55 


White piue 


25 


Ked pine 


30 to 40 


Pitch pine 


40 to 45 


Ash 


45 


Beech 


43 


Teak 


40 to 55 


Spanish mahogany 


40 to 50 


Stone, Brick &c. : — 




Limestone ... 


125 to 175 


Portland stone 


144 


Sandstone ... 


135 to 145 


Slate 


175 


Basalt 


187 


Granite 


170 


Masonry 


116 to 144 


Brickwork, ordinary 


112 


Concrete 


120 to 130 



INDEX. 



Annealing, 43 
Arch, 228 

— linear, 229 
Arched rib, 230 

— — hinged at ends and centre, 

231 

— — hinged at ends only, 233 

— — fixed at ends, 236 
Autographic stress-strain diagrams, 67 
Axial pull, 97 

Ayrton, Prof., 198 

Baker, Sir B., 54, 150 

Barba, 47 

Bauschinger, Prof., 53, 54, 73, 92, 93 

Beams, deflection of, 129 

— transverse bending of, 135 

— continuous, 143 

— resilience of, 136 

— stress in, 109 
Bending moment, 110 

— — diagrams, 113 

diagrams, how related 
to funicular poly- 
gon, 115 

— — and shearing force, 

how related, 122 

— of beams, 129 

— of long columns, 172 

— stress, 104 

Blows and shocks, effects of, 56 

Bollman truss, 169 

Bottomley, J. T., 41 

Bow, 158 

Bulk modulus, 13 

Cantilevers, deflection of, 131 

combined with beams, 1 19 



Cast-iron, test of, 31 

— crushing of, 50 

— data for, 90 

— beams, 113 
Catenary, common, 221 
Centrifugal whirling of shafts, 201 
Chain carrying loads, 218 

— parabolic, 220 

— inverted, 228 

— loaded with its own weight, 221 
Chaplin, Prof. W. S., 49 
Clapeyron's Theorem of Three Moments, 

146 
Christie's experiments on columns, 182 
Cold-shortness, 58 
Columns, 171 

— Euler's theory of, 172 

— Gordon's formula, 179 
Compression, failure by, 50 
Connecting rod treated as a strut with 

lateral load, 186 
Continuous beams, 143 

— — advantages of, 148 
Contraction of section in tensile tests, 44 
Counterbracing in bridge frames. 167 
Creeping in strain, 24 

Crushing strength, table of, 241 
Crystalline structure of metals, 46 
Curvature of beams, 129 

— — uniform, 130 

— — anticlastic, 186 
Cylinders subjected to internal pressure, 

204 

Deflection of beams, 181 

— — due to shear, i n 

Diagrams of strain and stress. 80 
autographio, BS 



244 



INDEX. 



Disc, revolving, 215 
Dunkerley, Prof., 203 



Greenhill, Prof., 200, 214 
Gun-barrel with initial stress, 212 



Ease, state of, 57 Hackney, W., 47 

Elastic compression of blocks, apparatus Hadfield, K. A., 95 

for measuring, 79 Hanging chain, 218 

Elastic moduluses, table of, 241 Hardening and tempering of steel, 44 

— strain, 10 — of metals after overstrain, 

Elasticity, imperfection of, 54 33, 42 

— — in cast-iron, 57 Heat, influence of, on recovery from 

Emery testing machine, 68 overstrain, 36 

Encastre beam, 150 Henrici, Prof., 158 

Euler's theory of columns, 172 Hodgkinson, 31, 50, 91, 179 

Ewing, J. A. , experiments on overstrain, Hooke's law, 11 

Hoop tension in cylinders, 204, 208, 211 

— in revolving ring, 214 

— — disc, 217 
Hysteresis, mechanical, 24, 55 

— — in overstrained 

steel, 39 



35 

— crystalline structure of 

metals, 46 

— extensometer, 75 
Extension, non-elastic, 45 

— percentage of, 47 
Extensometers, 29, 73 

— Bauschinger's, 73 

— Unwin's, 74 

— Ewing's, 75 

— — for compres- 

sion, 79 

Factor of safety, 27 
Fairbairn, 50, 52 
Fairbank's testing machine, 67 
Fatigue of metals, 52 

— — in elasticity, 55 
Fidler, Prof. C, 182 

Fink truss, 169 

Flow of solids, 33 

Forth Bridge, 150 

Fracture by tension or compression, 50 

— under successive blows, 56 
Frames, 154 

— perfect and imperfect, 155 

— redundant members in, 155 

— with semi-members, 156, 166 

— solution of by method of sec- 

tions, 156 

— solution of by method of reci- 

procal figures, 157 

— superposed, 169 

— effects of stiff joints in, 170 
Funicular polygon, 120, 162 

Gordon's formula for columns, 178 



Imperfection of elasticity, 54 
Instability of long columns, 171 
Internal stress, initial, 57 
Isotropic material, equations of strain 
in, 20 

Kelvin, Lord, 55, 68 

Kennedy, Prof., 29, 49 

Kirkaldy, D., 30, 43, 45, 48, 51, 65 

Launhardt, 54 
Linear arch, 229 
Linville truss, 156 

Martens, Prof., 58 
Maxwell's needle, 88 

— method of reciprocal figures, 

157 
Method of sections, 156 
Modulus of elasticity, Young's, 12, 16, 
73, 81, 94 

— cubic compressibility, 13, 16 

— rigidity, 13, 16, 85, 87 

— rupture, 107 
Moduluses of elasticity, relation between, 

14, 16 

— — table of, 241 
Moving loads on beams, 123 

Muir, J., experiments on overstrain, 
38 



INDEX. 



245 



Nickel steel, 95 

Overstrain, 25, 42 

— due to slips in crystals, 46 

— effect of in hardening metals, 

33, 35 

— in breaking down elasticity, 

35 

— experiments on, 35, 37, 38 

Pearson, Prof. Karl, 57 
Perfect frame, 155 
Permanent set, 10 
Perry, Prof., 184, 198 
Plasticity, 25 

— advantage of, 28 

— nature of in metals, 46 
Poisson's ratio, 12, 18 
Principal stresses, 4 

— — in a beam, 126 

Kankine's formula for columns, 179 
Reciprocal figures, 157, 159 
Redundant members in frames, 156 
Resilience, 14 

— of beams, 136 

— of shafts, 193 
Revolving disc, 215 

— ring, 214 
Rib, arched, 230 
Richards, E., 48 

Rigidity, modulus of, 13, 16, 85, 87 

Riveted joints, 49 

Rosenhain, W., crystalline structure of 

metals, 46 
Rupture, modulus of, 107 

Safety, factor of, 27 

Sections, method of, 156 

Semi-members, 156, 166 

Set, permanent, 10 

Shackles for testing machines, 66 

Shafts, torsion of, 187 

— centrifugal whirling of, 201 

— resilience of, 193 

— under end thrust and torsion, 

200 
Shearing strength, table of, 241 
force in beams, 110 

— diagrams of, 
113 



Shearing stress, simple, 4, 7, 21 

— — equality of in two di- 

rections, 9 

— — in beams, distribution 

of, 125 
Shear modulus, 13 

— in tension and compression tests, 

51 
Shells, 204 

— spherical, 207 

— of oval section, 207 
Simple bending, 104 
Single-lever testing machine, 59 
Slips in crystals of overstrained metal, 

46 
Slope and deflection in beams, 131 
Spangenberg, 54 
Spiral springs, 195 
Steel, stress-strain diagrams of, 30 

— data for, 92 

Stiffening girders in suspension bridges, 

223 
Strain, 10 

— elastic, 10 

— non-elastic, 24 

— — work done in, 14 

— simple, along one axis, 22 

— lateral, 12 

— hysteresis in, 55 
Strength, ultimate, 25 

— tables of, 239 
Stress, 2 

— intensity of, 2 

— distributed, 2, 96 

— normal, 3 

— tangential, 3 

— compressive, 4 

— tensile, 4 

— shearing, 4 

— principal, 4 

character of in simple push or 
pull, 5 

— fluid, 10 

— uniformly varying, W 

— in beams, 10'.) 
Stress-strain diagrams, BO 

— an'tographi* 

72 
Stretch modulus, 19 
Btromeyer, 58 

Struts, 171 



246 



INDEX. 



Struts with lateral load, 184 
Superposed frames, 169 
St Venant, 200 
Suspension-bridge chain, 220 

— — with stiffening girder, 

223 

Tables of strength and elasticity, 239 
Temperature, influence of on strength, 
57 
— influence of in producing 

recovery after over- 
strain, 36 
Tempering of steel, 44 
Tensile strength, table of, 239 

— tests, 29 
Testing machines, 29, 59 

— — calibration of, 63 

— — Werder's, 64 

— — Wicksteed's horizon- 

tal, 65 

— — diaphragm type, 67 

— — Emery's, 68 
Test-pieces, forms of, 47 

— long and short, comparative 

strength of, 49 
Theorem of three moments, 146 
Thick cylinder, 208 

— — exposed to external pres- 

sure, 211 

— — advantage of initial 

stress in, 211 
Thomasset testing machine, 68 
Thomson, James, 57 
Thurston, Prof., 58 
Time, influence of in testing, 41 
Torsion of shafts, 187 

— — in relation to power 

transmitted, 190 



Torsion of shafts beyond the elastic 
limit, 194 
— of non-circular shafts, 199 
Torsional oscillations, 87 
Tresca, 33 
Twisting combined with bending, 191 

Ultimate strength, 25 
Uniformly distributed stress, 96 

— varying stress, 98 

— — — analysis of, 

102 

— — — forming a 

couple, 101 
Unwin, Prof., 72, 91 

— — extensometer, 74 

Volume strain, 21 
Voussoirs of arch, 229 

Wade, Major, 65 

Warren girder, 156 

Webster, J. J., 58 

Weights of materials, table of, 242 

Werder testing machine, 64 

Weyrauch, 54 

Whirling of shafts, 201 

Wicksteed, J. H., 59, 65, 72 

Wire guns, 214 

Wohler's experiments on the fatigue of 

metals, 52 
Work done in straining, 14 
Wrought-iron, data for, 91 

Yield-point, 29 
Young's modulus, 12 

— — measurement of, 73, 

81, 83 



CAMBRIDGE : PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS. 



* 56 








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